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Where would modern bridge players draw the line in the following paradigm?
Take a hand like ♠ AK ♡ K ♢Txxxx ♣ Txxxx. In the old days (about 100 years ago), some experts might have evaluated the hand at 2.5 quick tricks and two five card suits and opened it 1 diamond (or 1 club). I doubt that modern experts would do this, since it only has 10 high card points.
Improve the hand slightly to ♠ AK ♡ K ♢Jxxxx ♣ Jxxxx. Now it (technically) has 12 high card points, as well as the two five card suits. Would modern experts open this hand?
If not, how much would the hand have to be improved? Perhaps to ♠ AK ♡ K ♢Qxxxx ♣ Jxxxx? Or might the key improvement be to ♠ AK ♡ Kx ♢Jxxxx ♣ Jxxx, shortening the club suit but protecting the formerly singleton K of hearts, pushing its value closer to "3"?.

Based on Mycroft's answer below, it appears that the key improvement for the 12 point hand might be protecting the king of hearts so that it actually is half a quick trick.

That's still horrible.  But it might get opened by the "all 12s that aren't 11s" crowd.  I would be happy to pass this dreck even so, and see if the opponents play the misfit with all the top honours in short suits.  Playing a strong NT, I am not as conservative as I am playing a weak NT (where 1m openers need to "say something"), but I still am more willing to pass sub-minimums if I'm opening a minor.

Two statements here, before it goes into the weeds:

One of my rules (which nobody I trust disagrees with) is "Once you have decided your hand is an X, it remains an X no matter what happens.  It could be a good X, a bad X, and incredible X or a horrible X, but it never becomes not-X."  X can be "opening bid", "game force", "single raise", "balanced hand", whatever.  Here, we're talking "opening bid".
Quick tricks aren't what your question states they are.  Your example hands have 2 only QT, until your last one which is 2.5 (which I will admit is "closer to 3" than 2).  Quick tricks are (quote from Wikipedia, my emphasis, but many others online):

2 quick tricks = AK of the same suit
1 1/2 quick tricks = AQ in the same suit
1 quick trick = A
1 quick trick = KQ in the same suit
1/2 quick trick = Kx (not K singleton)

This is because a singleton K will never take a "quick trick".  It needs the A or Q or it's as good as the 2.  Kx will take a trick half the time depending on where the Ace is.
Now:
Anybody who required 2.5 QT (or, like K/S and Goren, stated that 2.5 QT was a mandatory opener, could open 1.5 or 2 if...) would never open these hands as they're awful, 10 Milton HCP or no.  Even 12.  Even in 1940, minors got outbid by majors, there are no extras, most HCP are outside the suits, the spades are inflexible,...
The switch to Milton Work Count doesn't change the analysis.  Remember, Miltons don't take tricks; they don't win partscore battles; they're just a convenient way to start the analysis of a hand.  In discussions, people will start with (unless they just give the whole hand) "I had a [shape] [hcp] count, with..." but the "with" is the important part.
What that "with" is, depends on known tools, skill at using them, and need to use them.  Using LTC as a secondary tool, "I had AK K 55, so 7 losers, but my suits are minors".  Using "in-out", "I had AK K 55, my 'out' honours are good, but there's nothing 'in'."  Or you might get the kind of analysis I used above.  Or...
All this to say that "the switch from QT-based primary evaluation to HCP-based primary evaluation" just would not be any reason to change whether the hand is opened or not.  If "modern experts" don't open this hand, it wouldn't be because it "only has 10 HCP" (see AK843 K8432 8 64, for instance, probably an auto-open), it's because your example hands are too awful to commit to "being an opener".
But having said that, they might be opened now - for two reasons:

Openings are just getting lighter and lighter.  In standard, "decent 11s and all 12" is pretty common, and at least you have a plan (1♦, then rebid clubs).  Less likely the 10 (effective 7) counts would be, but the 12s might - but they'd be "awful openers", at least until partner shows a huge fit for one minor (and even then, I'd expect my last decision would be "do I sacrifice?")  But if they are opened, it's because partner knows they can be opened and caters to (or has the system cater to) them.
Many players don't play a Standard system, and do so so that they can open lighter yet.  Modern Precision players open "10-15", especially NV, and would open many "9s that look like 10s".  They might open 1♦ on those hands (having a safe 3-if-necessary-♣ rebid if the auction goes competitive).

How much would you have to change it to get a majority in flight A to open?  At least moving the ♥K into a minor.  Or switching one of the suits to spades.  Or make it a real 13 (QJ in a minor?)  Not sure.
It's good to notice these hands are awful, and that single-evaluation methods fail on some hands, QT, HCP, LTC, ZAR or whatever.  But know that experts (modern or otherwise) never use a single method of evaluation in their analysis, and never make a judgement based on a single factor (okay, "never bid slam with two quick losers"; "never" never means never in bridge).  You shouldn't either.

I think you're asking for a single rule for a question which is multi-faceted (as illustrated by Mycroft's extensive answer).
After 'Open all 13 points hands', the choices get contextual below 13 points. Obviously that doesn't mean never open beneath 13, but one shifts from the 'I am strong enough to open' path to one of many 'My hand has shape-merit' bids. Now one looks for Offensive-strength backed bids matched to 'reasons' for bidding: possible ruffing potential/part-scores/sacrifice/lead-direction/pre-empt, etc.
The opening hands (above) are bad because of their low offence to defence ratio (ODR). They're for defending rather than playing and no reason to open. (You'd just need Partner with A in Clubs or Diamonds (67%) to defeat opponent's 4H, while a club void with D-AKxxxx for 6D quite unlikely and you'd discourage partner's desirable spade lead by bidding diamonds). Conversely, when partner's strong they've a bid of their own, so you never miss a game or slam.
I'd say it wasn't possible to evaluate many 12- point hands against a rule, or find a line, but better to measure Offensive/defensive merit (trick-taking & losing potential) in Context (vulnerability, seat, reason-for-bidding). Experts judge those biddable low point-count openers from feature more than formula although feature-based evaluations are varied -  Where points are in long suits? Are short suits a strength or weakness? What can opponents make? Bid or Defend? etc.


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Am I underestimating the differences between these two hands if I bid them the same way?
In the first case, I was in "third" position, only them vulnerable at match points. Here I pre-empted 3 diamonds with the following 13 high card point hand: ♠ 2 ♡ KT ♢ AKQ6432 ♣ J53. I would not have done in the first or second position, but opposite a passed partner, I didn't see much of a chance of missing a slam. But it raised eyebrows with others at the table because I had 13 high card points.
The following hand, which I had played a few days earlier, was more obviously pre-empt material:  ♠ A86 ♡ 2 ♢ AQ987432 ♣ 3. But I opened it 1 diamond instead of 4 diamonds in first position (neither vulnerable) to give my partner a chance to speak. The opponents found their heart fit, bid and made five hearts giving us a bottom.
I feel that the two hands are similar enough to bid the same way, and that either "pre-empt" or "one diamond" could be the right bid depending on position, vulnerability, and other factors. Putting aside the fact that the first hand has 13  (Milton) "Work" points, and the second one, "only" 10. I consider the first hand to have 7.5 potential winners, and the ability to make 9-10 tricks if partner has his statistical average of 9 "Work" points. I believe the second hand may have 8 potential winners because the eighth diamond compensates for the lack of the king, and similar potential of making a 3-4 level contract. And I believe that both hands have only two defensive tricks (the ace of diamonds and a side honor).
Am I wrong to consider the two hands equivalent and bid them the same way, or are there major differences that I have overlooked?

In all my non-Namyats playing partnerships, I'm opening the first hand 3N in 3rd seat.

@AlexanderWoo: I have recently "come around" to pre-empting on this hand for a reason that surprised me. That is, with seven playing tricks, I can "bid to make" with seven tricks in hand, and a reasonable hope that partner will have the two I need, (instead of expecting to go down one or more). If I have 13 hcps, the average of the other three players is nine. And if partner doesn't have two tricks, the opponents might have a game that the pre-empt might prevent.

I have narrowed the question by focusing only on my own thinking (and its possible flaws, rather than what "people" generally think.

Short answers: Yes.  Yes.  Yes.  Yes. That's what makes bridge interesting.
There is no right answer(*).  It depends on your partner, your partnership, your agreed system, your opponents, the state of your game, the field, and your level of caffeination.  Among others.
Please remember there is exactly one person whose opinion matters here. You say you got "raised eyebrows" from the table.  If it was the opponents, their opinion Does Not Matter - especially if the preempt worked.  If it was partner...?

Is this a new partner?  Maybe stick to stuff she will understand.
Is this a partner that will preempt aggressively (or open aggressively)?  Then the chance that you will win an auction starting at the 1-level is lower than if partner is sound and could have (say) QJ9xxx in one of the majors or have a flat 12 count.  So, better to bid at the 3 level(**).
Does your partner try to blame you for all your bad results?  Then do the thing that will Win The Post-Mortem(***).  Sure, it may not win the board, but both you and partner are more likely to be concentrating on the next hand when it comes rather than the argument you get into on this one.
Is this a pair you expect to beat playing with the field?  Then make the field bid.  Similarly, if in a team game you think you're better than them, why open yourself up to a "random" 6-10 IMP swing with an unusual bid?
Is RHO a "can't be preempted" player?  Maybe now's the time to bid 3 and get them too high with 20-20.
Is RHO a player that really pays attention to the auction? Again, a 3-bid with the 13-count, or a 1-bid with the 10-count, might elicit a misplay.  At least, if you're known to be a "straight actor", anyway.
Are the opponents frisky with the red card?  Maybe it's time to walk the dog (bid 1, then 2, then 3, hoping to eventually get doubled) with the likely 8 tricks, and go for +530 (or -100 into -140 or -300 into -620). Especially if your room is full of people who will re-re-bid bad suits just because they're long.
Are your opponents the best pair in the room?  Then they're likely to beat you anyway, but even they might not be able to find the 30-point-deck slam if you open 3.  If the opponents are weaker - not really much weaker, frankly - it might be harder to find the same 30-point-deck slam opposite a "real opener".
Is this a regular partnership?  Then you should have discussed this already; do what partner wants you to do (including "use your own judgement").  Is this a pickup?  Well, usually in your local community, you have a good guess as to how they think from how they play against you - but if the information's balanced, it might be better to "play straight".
Are you running a 45% game?  Maybe it's time to try something unusual to get you to 50%.  Is 42% going to get partner yelling at you?  Maybe it's time to play straight and take your 45%.  Similarly, are you running a 62% game?  Do you want to increase randomness by taking an uncommon action?  Or hope you can ride your good game with your better than average luck/skill?
Re: suit - that's always a consideration.  That's (one of the reasons) why requirements for 3♣ and 3♦ openers are frequently different than 3♥ and 3♠ openers.  In third seat, it's more interesting, though, because the other big reasons are "what does partner need to bid 4?" and "what should partner have to sacrifice (especially 4♠ over 4♥)?" which might actually give you more flexibility in minor preempts than major ones.

All of this (and more, I'm sure) to say, the primary answer is "yes, you can use judgement to make an off-system call, or even an off-normal call.  In fact, doing it at the right times more often than others is one of the things that makes a better player. Third seat favourable is one of the 'freest' times - good for thinking about it."  Do I think your judgement is right on these hands?  Well, my personal belief is that the overstrong preempt in this position is more likely to work in your favour than the upgraded 1-opener.  But there's never a guarantee, and I am not Larry Cohen.
The secondary answer is "if you annoy or confuse your partner, you will not score well, and you are less likely to be asked again." To me, that's more important.
The tertiary answer is "this session (like all sessions) is an audition." If you want partners in future that accept some off-book judgements, then play that way.  If you want partners who make The Right Call, then do that.  Play the way you want to be thought of by the person who might have a game free next week, both technically and in the way you treat your partner, and they might ask you sometime.
(*) There almost never is "the Right Answer" in bridge (at least, in the auction).  Again, that's what makes bridge interesting.  "I am right because The Book says so" died with Goren; unless you have agreed with partner/the rubber club rules/the "one system" event to Play By The Book.
Similarly, one swallow does not make a summer.  Just because this hand got a bad result, doesn't mean the judgement was bad (or vice versa).  There are people out there that look at the result on each hand and work out how best to get to the winning score; these players are called "Resulters", and they either burn through partners not willing to be wrong again (even if it would have been right 70% of the time) or end up with a system so full of exceptions and convolutions that they can't bid the 60% of the "normal" hands correctly.
(**) or 3NT with the first one, if you play Gambling.  Yes, "zero outside controls", but partner with 2 aces and out won't bid 3NT after 3♦ either.  And if partner's also on the risk-taking side (and you're not going to have a problem if she guesses wrong), she might pass with "nothing" and hope to go -250 or -300 into game.
(***) Do Not actually try to win the post-mortem.  With pickups, avoid having a post-mortem beyond "do you want this call with my hand?" or "did we agree to play one-way or two-way Drury?"  With a (want-to-be) regular partner, the post-mortem should not be something to win or lose, rather something to get on the same page over, or decide that something needs to change, or work out how better to get a message across in a way that both players can understand.
The thing here is that the kinds of partners who look for ways to make it "your fault" are less likely to have something to hang on to if you play it straight.  So play it straight and avoid the hassle.

It must be nice to be able to play enough Bridge to be confident as to whether you're currently (!) "running a 45% game" versus slightly more or less.

It is, yes.  But "estimating your game" is a critical skill in duplicate, given that unlike most other games, you don't get an up-to-date running score.  I have a Mike Lawrence book - "play a swiss teams with..."  Hand 1, he ends with "as I expect they will bid game, I estimate a 10 IMP loss."  And he keeps a running estimate for the other 55 boards.  And compares his estimate to the "real result" at the end of each match.  It's that key a skill.  If you aren't there yet, try explicitly ranking your scores as you write them, on a 0-4 scale. See how close you get, after 3 or 4 sessions.

With the edits and the change in focus, my other answer (while still valid, and I will refer most of my arguments for what I state here, there) looks like it's a little odd.  Mostly because it assumes the answer to the current question asked is understood.  So, let's look at that.

Am I wrong to consider the two hands equivalent

Oh yes.   The two hands are very different.

One of them will take 9 tricks in diamonds opposite effectively nothing; and 9 tricks in 3NT if partner has something in the black suits.  It is also likely that they can make 4 of either major, especially if partner has three diamonds - and at that point, 5♦ is a great sacrifice at these colours.
The other one needs a lot of help to take tricks on offense.  Sure, partner's singleton ♦K is "a lot of help", but without it you could lose 3 diamonds and the four outside cards if partner's 8-count is in the round suits.  Or if partner's spades are such that you only have one spade loser, that's 300 into likely to be no game.  Even 100 (undoubled) into no game is likely to be poor.

But let's be less concerned about bad fits, and say that they can make game.  Now partner's sacrifice could be 800 or 1100!

The additional difference between the two hands is the second "sure" trick of the ♠A vs the "half-trick" (less than that, actually, as if they do have points for game, LHO's hand will be significantly stronger than RHO's, so more likely to have the A) of the ♥K.  So it's more likely that "some cards" will hold them to 9 tricks.  Now partner's sacrifice is only 500, but it's a phantom.

But:

and bid them the same way,

Absolutely.  If they both fit your agreements for a third-seat pre-empt, then you could bid them both as a pre-empt.  And that agreement isn't wrong; the description of one partner's third seat weak 2s is "she has 13 cards, some of which (at least 5) are hearts." Partner will take that into account when deciding what to do - which may lead to less-than-optimal results.  That's the downside to go with the upside of "good luck lefty, you're at the 3 level" more often.
You can also agree that in third seat, you can take liberties with your 1 bids.  In a standard framework, there are downsides to that, as partner has to "not hang" you when you step out, which makes it more difficult to have good auctions when you actually have the goods, but it can be very effective, especially in a weaker field.  As implied in the other answer, it does tend to be more effective in the majors than diamonds, though.
Why "a standard framework"?  I play Precision, and I play Precision so that I can open good 10 counts (even in first seat) and partner won't hang me.  I can bid that solid suit 1♦, and if I get a second chance bid 3♦ to show "tricks and a long suit", because I'm limited to 15 HCP.  I can rebid 2♦ with the broken suit (and maybe bid 3 later) showing that hand.  Or maybe I can't; see other answer for "it's less likely to work".

are there major differences that I have overlooked?

Absolutely.  See 1.
Are the differences sufficient to mean that it's wrong to make the same call third seat favourable with them both?  See the other answer. Summary: up to what partner will expect, and what "stepping out" actions partner will forgive you for, and how willing you are to take the blame for a bad result when you "step out", no matter what partner did after.
For those saying "but what about pickups, where you don't know any of that?":

As in my other answer, it's only very rarely you have a "true pickup" without knowledge from playing against them, or playing with common partners, or even from knowing how "people in the area" are.
as a club Director, I've played "pickup" (to fill in a table, or as the spare) more than most; and when I do, I rarely have time to discuss system well (I frequently describe our system as "we started discussing 5 minutes after game time.  We've agreed...")  Even then, I get three things, and one of them is "what's your preempt style?" (the other two are "what do we play against their 1NT", and "what answers to 4NT?") If you do not know this - even with a pure pickup - you can't judge these hands correctly for your partner, and they can't judge what to do after.  So find out beforehand.

"and 9 tricks in 3NT if partner has something in the black suits." - if partner only has a club stopper, surely it won't be hard for opponents to find spades?

I did very carefully say "suits".  It's actually more likely to make if the only stopper is in the spade suit, for the reason you mention.

Ah, I naturally understand "something in the black suits" as meaning either, not necessarily both. But I guess you would have said "something in a black suit".

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Should the responder to a strong two club bid have been given a "double negative" option?
At matchpoints, with only them vulnerable, I was defending against a strong two club sequence bid by a husband and wife team.
Wife opened 2 clubs (strong) on my right with ♠ AK64 ♡AQ ♢AQ ♣ KJ742. Husband bid 2 diamonds (waiting) after my pass.
If opener rebids 2 of a major after these two bids, a responder who has a bust can plead "double negative" with 2NT, which allows opener to rebid only three of the major (that is less than game). If opener rebids 2NT, responder can pass at that point to invoke the double negative. At least that's the way we play in our club.
But wife bid 3 clubs. Husband was supposed to bid on, even with only ♠ 97 ♡97642 ♢J962 ♣ Q9. But he passed, and the couple got a top because they made 3 clubs and all the other players with their cards went down at 3NT.
Should wife have limited her re-bid to 2NT and given her husband the option of passing (and earning the top)? I don't see the point of her bidding three clubs with her hand. (If I plan to rebid three clubs after a strong two, I would want extra strength, i.e. six clubs to go with my 22 high card points, or 25 high card points if I only had five, because it takes 11 tricks to make a club game. And the three club bid forecloses responder's 2NT "double negative" option.)

Everyone I know plays 3C rather than 2N as the double negative..  In this case, are you sure they play 2D as waiting rather than as strictly negative?  (But, yes, I rebid 2N with that hand, and with AKxx A AQx KQJxx I open 1C, not 2C)

@AlexanderWoo: In our club, 2d is "waiting," not necessarily negative, and 2NT is the second negative. A 3 club bid precludes this double negative and is forcing. 2NT (by opener) is not.

The standard second bid double negative in North America is 3C over 2H/S, 3D over 3C, and 3H over 3D.  Many players now prefer an immediate double negative, where a 2D response promises an ace, a king, QJ suited, or 2 queens, and one responds to 2C with 2H with less than that (and responds 2N to 2C with a heart positive)  There are different arguments as to which of those two is better, but they are definitely both better than 2N double negative, because 2N double negative will wrongside many contracts.

@AlexanderWoo in classic Acol 2NT is the second negative. But we have established elsewhere that Acol may well have moved on from when I learned it.

I'm going to say the quiet bit out loud: (1) This partnership has serious ethical issues. (2) It's hardly relevant, because they get 3+ bottoms for every 2 tops and rarely finish above a 45% game, never mind a 50% game. (3) In consequence, analyzing their bidding is a fools' errand .

@ForgetIwaseverhere: You are right. The majority of the couple's scores are 45% or lower. The wife does better when not partnered with husband.

I agree that opener should have rebid 2NT after 2C-2D. If she wanted to show a strong club-spade two-suiter, she should have opened one club and shown the spades with a game-forcing rebid (eg 1C-1H-2S). That would have been a reasonable approach with e.g AKJx AQX x AKJxx. With her actual hand, with mediocre clubs, semi-balanced shape, and values in her short suits, 2C then 2NT is the best description.
Second, I agree that opener's actual sequence was forcing and should show better strength and/or shape than she actually had.
Finally, I disagree that 2NT would be negative after 2C-2D-2 of a major. Typically 3C is used as the double negative rebid. 2NT is forcing for one round and denies a 3-card fit for opener's major. Many pairs will have discussions that differentiate hands that rebid 2NT vs 3NT. It is losing bridge for 2NT to show a negative hand, because 3NT is one of the most likely landing spots after 2C-2D-2M.
It is also perfectly reasonable to play without a double negative bid after a 2C opener. In these auctions, defenders have few clues from the auction when making their opening lead; your best chance at a good score may be to land in a thin game that makes through misdefense. Moreover, if you only open 2C when balanced, single-suited, or have game in hand (preferring to open at the one level with two-suiters without game in hand), the double negative from partner is rarely relevant and takes away space that may be used more profitably to find the right game or slam.

I agree with pretty much everything ruds said. Just adding a few pieces too long to fit into a comment.

I think that in very distant past 2NT would have been the second negative after 2C-2D-2Major. This is on its way out for the reasons described in ruds's answer.
Search for a convention name cheaper minor. This is used as a
double negative in sequences starting with 2C. The idea is exactly to
avoid wrong-siding NT contracts. This usually won't consume too much bidding space, but there is the problem of how to show the double negative after 2C-2D-3D. That gives a headache even for very good players. See a poll at Bridgewinners.

Alexander Woo mentioned the possibility of using 2C-2H as an immediate double negative (when 2C-2D is waiting, semipositive, and always forcing to game). I play this with my regular partner, but we have extensive system notes. That is necessary because:

The responder (to a 2C opening) needs a way of showing a positive hand with 5+hearts. Many players use 2C-2NT for that.
But you also need to find ways for the opener to describe their hand now that the 2H rebid has been taken from them! Or rather, the opener need not describe much, they simply need to ask whether the bust partner has any support (lengthwise) to the 2C-openers major (or the minor on very rare occasions)!! Realizing this my partner came up with the idea of using 2C-2H(bust)-2S as a bid asking primarily about major suit lengths. Still, we ended up agreeing on requiring a 2C opening to be a bit stronger than what the average local club players do. For example, we use 2NT for opening balanced hands in the 22-23 range (we already had some strong balanced hands in our 1C opening). Anyway, my point here is that doing this requires extensive discussions for relatively rare gains.

The last thing I wanted to say is not to worry about your opps getting a good score on this board. They were very lucky. Change the club 7 into a tenner, and 3NT is more attractive (not a gimme by any means, but better, because declarer can then force an entry to the dummy in clubs). Also, move one of the opener's low spades into hearts, and 4H is beginning to look attractive. A fluke, I think. If someone with insider information knows the pair to have a history of ethically dubious actions, do check with the TD. In my opinion this board alone is insufficient evidence of anything like that.

The husband is just a bad bidder. The wife was the TD. She told her husband, "You weren't supposed to pass my 3 clubs bid.

LOL! Thanks for the explanation, @Tom.

Actually, the possible simple systems after a 2C opening were discussed in this old thread.

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Did I have enough information to call four spades in this situation?
With only them vulnerable, partner open 1 club (possibly a "convenient" minor) in second seat. My right hand opponent overcalled four hearts (showing eight). I passed with the following.
♠ AT8652 ♡2 ♢93 ♣ AQJ7.
They made it, giving us an average minus score. Others did better to bid four spades, and after I worked this out at home, I felt that I probably had enough information to do the same. My premises include the following:

I had 11 high card points; partner had at least 12-13 to open, so we had at least 23-24.
The opponents had the bulk of high card points in hearts. Giving them 10 (they actually had 9 because partner had the singleton jack) meant that they would have something like one ace and one king at most in the remaining suits.
The opponents would probably have 10 hearts between them (they actually had 11), leaving us three, meaning we would have 23 cards in the three remaining suits.
Given 3, above, our worst possible distribution in the three non-heart suits would be 8-8-7, giving us at least one eight card fit between spades and clubs. If our short suit were diamonds (likely, since I was short), there would be a good chance of a double fit. This, in fact, was the case.
Given my singleton, we had one loser in hearts, and two more losers from the side ace and king attributed to the opponents under 2, above, meaning we probably had at least ten tricks. Put another way, we should have at least nine tricks with an eight card spade fit and 23 or so high card points. Given a double fit in the black suits (and resulting shortness in the red), the chances for a tenth trick seem excellent.

Is the above analysis sound, or was it reasonable to pass because of hidden flaws in my reasoning?

You don't really know how many points opponent has. A 4H bid there could reveal blank on any other suit but hearts. It's a destructive bid from the old days actually intending to go down rather than permit opponent to get game by normal bidding. (I learned on an 80 year old book.) It could also turn out to be slam material by opponents.

Yes, you are reasoning along the right lines. But there is a shortcut you can use here: you have an opening hand and partner opened the bidding. You cannot allow the auction to end without making some call other than pass. Partner has no idea that this is your side's hand, not the opponents'. So now the question is, which call?
Double is the most flexible action. It's needed for hands that don't have a strong opinion about strain, or even whether to defend or declare. This isn't that type of hand. You have a very offensive hand. You are short in their suit and all your honors are in your long suits. You should bid your 6-card suit. It won't always work out (eg if partner is 6-4 in the minors with a stiff spade, you'd rather be in 5C), but that's why opponents preempted. It'll work out better than pass almost every time.

Although you only have 11 high-card points, they are all in your long suits. Additionally you have a fit in clubs with partner and very likely a fit in spades also.
As ruds points out, this hand belongs to your side. Bid 4S. Even if partner hadn't opened, you are odds-on to be no more than two down, a great sacrifice against an opposing game. As partner has bid, you would have to be very unlucky not to make it.
Why did partner open 1C? If you are playing some form of Standard American, then partner either has an unbalanced hand with long clubs, or a balanced hand with 12-14 high-card points. If the former, then you have a spade fit. And if the latter, then he can retreat to 5C in safety, due to your support.
Finally, given the eight card heart suit, partner knows you are short in hearts. A double from you would have to show a double based on values, not trump length. Bidding spades informs partner that you have a suit good enough to compete at the four level, not just scattered values.

You need to bid 4S.  Keep in mind that there are TWO ways 4S can be a good bid.  One is that it makes.  The second is that it is a good sacrifice against 4H.  Don't neglect the second possibility.  Try to come up with reasonable holdings around the table where neither 4S nor 4H will make; it's not impossible but you'll find it pretty hard.
Imagine your partner with Kxxx xxx A Kxxxx (which they probably wouldn't even open) and both 6S and 4H are making!

It was Marty Bergen who said something like, "bid if the cards that you need to make the contract can be contained within the minimum of partner's previous bidding." Here you have three probable losers in the red suits. But you can make ten tricks in the black suits if partner has the right cards, specifically the KQ of spades, and K of clubs,
A four spade bid would be right even if partner was in third seat and opened "light" with 9-10 points, because the above needed cards can be contained within this (hypothetical) minimum. Here, partner's second seat opening shows at least 12-13, more than enough.
Even if partner didn't have all of the right cards, you'd be an underdog only if she had the most useless possible cards for her bid, something like the singleton king of hearts (dropped) QJ of diamonds ("stranded"), the Q of spades, and one, but not both of the black suit kings. You would even be a favorite to make your contract with a slightly more heavy (but more useful) red suit holding for partner: e.g. Jxxx K KQx Kxxxx. Your one diamond trick would offset the loss of a black suit trick.


text:
Can I reasonably expect my partner to have a four card suit when responding to a takeout double?
With only them vulnerable at matchpoints, right hand opponent opened (a weak) two hearts in third seat. I doubled (for takeout) with the following:
♠ AKT
♡ Q
♢ AKQ642
♣ A93
After left hand opponent passed, partner bid two spades. I put him on "four and four" (four high card points and four spades) and raised him to game. I counted nine tricks in my hand (given my long diamond suit) and hoped for one from him..
Partner actually had "eight and three" (eight high card points but only three spades. We went down because his hand was:
♠ QJ4
♡ A6
♢ T875
♣ JT65
I would have expected him to bid one of the four card minors, in which case I would have bid diamonds. Conversely, if he did bid spades, I would have expected four (unless his distribution was 3-4-3-3).
Did I have the right to expect that, or did he reasonably bid his three card spade suit? Was it correct to try for the higher-scoring major suit game at matchpoints, and treat my minor suit as a side suit, even though that didn't work in this particular instance?

I think you were both mistaken in this auction.
First: there are a few hand types that would cause me to bid 2S with only 3 on this auction, but this is not one of them. With 8 points, the three level isn't dangerous, so just responder should just bid their hand with 3C. Perhaps without the HA I might bid 2S - now 9 tricks look like a lot, especially if partner has only 3 clubs. With Kxx x 9xxxx JTxx I might also risk 2S at match points, fearing a forcing defense against 3D.
As to your hand, why jump to 4S? Even if partner has four spades it may not be the best contract. Imagine Jxxx KTxx x Kxxx (example 1). 4S is dicey but 3NT is a piece of cake. Alternatively, what about Qxxx xxx Jx QJxx (example 2)? Now 4S essentially requires 3-3 trumps (less likely than usual because of the preempt) and a little luck, while 5D is cold (and 6D has chances).
Your hand is right on the border between a 3D call and a 3H call. I think I would choose 3D, but change the HQ to a small one and a club spot to the CQ and I would bid 3H. With the actual hand or example 1 above, you'll get 3NT out of partner with either 3D or 3H. Example 2 is a bit trickier. If you bid 3H, responder should bid 4C, and then they'll raise your 4D to game. If you bid 3D, partner has a tough choice. Over either call, a partner with 5S and no second suit can rebid their spades.

"Your hand is right on the border between a 3D call and a 3H call". I'm not familiar - what exactly does a 3H cuebid promise (or demand) in this situation?

@KarlKnechtel It shows a very strong hand; I would say that it is forcing to game in this auction. Over a minimum bid by advancer (as here), it may indicate a slam try in partner's suit or a game-forcing hand with a suit of its own, or a game-forcing balanced hand without a stopper in opponent's suit. I would say that advancer's first priority is to bid no trump with a stopper. With extras and a 5-card suit, jump rebid. Introduce a 4-card suit, or rebid your suit with nothing else to show.

A further point is the alternatives to Double when looking for 5 spades. This hand is ideal for 3H asking for a  stopper. A or Kx is exactly what you need with 9 probable tricks  (note: leaping Michaels generally applies after weak 2's. Alternatively using lebensohl-over-weak-2's, an immediate 3NT also suggests 3NT while denying a stopper). Without a stopper and 5 spades then 3S makes a good response over 3H, and any other reply points to 5D (even 6D maybe!).

I'll add to @ruds fine answer that I would bid 2 NT with Advancer's hand in preference to 1 Spade. That shows a balanced hand with 8-10 HCP, no 4 card spade suit, and no five card minor suit; exactly what was held.
Further, Advancer should not bid a 3 card spade suit unless a notrump contract is feared. Here that is definitely not the case - rather it looks likely to be the best landing.

Good point! I completely agree.

@postrationalist: I don't understand the point being made by your comment. Did you perhaps intend the comment to be to the original question? It seems a complete non sequitur to my answer.


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Under what cirumstances does a negative double call for a NT response?
At matchpoints with neither vulnerable, and a new partner, I opened one club in second position with ♠QJ75 ♥AJ4 ♦4 ♣AJ964. Left hand opponent overcalled one spade. Partner doubled (negative) with ♠3 ♥K872 ♦Q53 ♣KQ753.
I interpret a negative double as "I don't like your suit, and I don't fancy defending against the opponent's overcall, so please choose between the remaining two (red) suits." I bid two hearts to show my three cards in that suit.
The best contract was 4 clubs, but I didn't know about partner's club holding, although she knew "something" about mine. Given her bidding, I would have expected her to hold something more like ♠3 ♥K872 ♦KQ753 ♣Q53.
She said that she thought of her club holding as the backbone of a no trumper, and wanted me to bid one NT instead of two hearts. (We got a bottom for going down at a seven card heart fit, while others made 2NT, (but not 3NT.)
Was 1NT or 2NT the correct bid at any point in my bidding sequence? Or did I reasonably expect better disclosure of her club holdings?

In modern bidding (according to wikipedia, since the 1980s), negative doubles are oriented entirely toward the unbid majors, and don't promise anything in particular about holdings in the minors. Negative doubles at the 1-level (or even the 2-level in many cases) don't show particularly offense-oriented hands. They may be perfectly happy to defend if opponents come back in. Generally when we're bidding at the 1-level, we're not yet particularly considering offense-defense ratio, because our partnership hasn't had a chance yet to exchange enough information to make a good decision about declaring vs defending.
In this case, when you open a minor, LHO overcalls one spade, and partner doubles, the entire message is around the heart suit. It shows 4 hearts and enough values to force to the two level (probably something like 6 or 7 at a minimum) with no upper limit, OR 5+ hearts without the values to bid 2H (so limited to something like 6-9 points). Partner may have 5-card support for your minor or a void; likewise, they may have length in the unbid minor or not.
Since you don't know you have an 8-card heart fit, you shouldn't advertise to partner that you do. Because the negative double isn't a distributional bid just showing hearts and diamonds, you don't need to stretch to show one of those suits. Here, you're very happy to bid 1NT, so you should do that. This Larry Cohen article describes standard rebids after a negative double; note that the 1NT bid doesn't even necessarily promise a stopper in opponent's suit! By bidding 1NT, you're not giving up on hearts. If partner has a weak hand with long hearts, they can sign off in 2H.

I should say that 2C is also an acceptable bid here if you're uncomfortable bidding 1NT with a stiff diamond. The point is that 2H is not the right bid.

Why are you showing a (non-existent) fourth heart rather than showing your (actually existent) fifth club. Just rebid 2 clubs and stop trying to Hideous Hog your partner.
Re comments below:

While partner is entitled to assume four clubs in your hand when you don't rebid notrump, you need a bid available to show the fifth - especially when it's a decent suit, as here. That is done on this hand with a second round rebid of 2 clubs. Partner has no basis to assume that this shows a sixth club.
I know that a 1 notrump rebid is being floated around this thread, but I can't agree. Partner is entitled to assume that this shows a balanced hand of specifically 12-14 HCP, and despite your good spade stopper I believe that showing the fifth club first is much more important because of the stiff diamond. You really don't want to be playing notrump until you have good assurance that Partner has at least 3 diamonds, and preferably 4, with a holding that opponents cannot easily lead into without giving your side a second trick. You know both opponents have length in this unbid suit, and hence it's coming down at opening lead unless from a holding that is compromised by doing so. Further, you want it played from partner's side and not yours.
So if notrump is the right spot: get there by rebidding 2 clubs first (limiting your range and showing the club length), and then consider cue-bidding spades to show the strong stopper for notrump.

It may be a "club: thing. When people in my club make a demand bid, they expect you to rebid a three card suit. In this context, a rebid of two clubs would imply six, not five. Even so, maybe two clubs is a smaller "lie" than two hearts.

Someone suggested that I rebid 1NT instead of 2 hearts.

I would say this depends on your partnership philosophy about 3 card raises of responder's suit.  Some partnerships frequently raise on 3 card support, risking playing a 4-3 fit; some do it rarely or never.
You should treat the double as a bid of 1H and act accordingly, except that you now can't bid 1S.
A few expert partnerships actually raise 1H to 2H with your hand, even with the 1S rebid available.  I think that's fairly uncommon but not unheard of.
Let's say the bidding went 1C-1S (opps silent), and you hold AJx QJxx x AJxxx.  You're clearly too weak to bid 2H.  Do you bid 1N, 2C, or 2S?
Many (particularly North American) experts take it further.  Over 1C-1S, they'll be bidding 2S with something like AJx QJxx xx AJxx, even with a perfectly good 1N rebid available.  Or they might bid 2S with AJx QJx x AJxxxx with a perfectly good 2C rebid available.
Now - the partnerships that frequently raise responder on 3 card support usually also play conventions (such as some form of spiral raise) to diagnose the 3 card support if responder is interested in game.
How common opener raises responder on 3 will also impact how the partnership reacts to further competition from the opponents.
This being a similar situation, I would suggest for ease of memory to treat it the same.  (Note that, if advancer bids, I do play support doubles in this situation.)
(Personally my preference is to almost never raise on 3 cards, but I would bid 2S after 1C-1S on AJx QJxx x AJxxx (but 2C with Jxx AJxx x AQJxx and 1N with Jxx QJxx A AJxxx (if not playing weak NT)).  Partner is allowed to insist otherwise though.)
Of course you'll have to discuss with partner your partnership philosophy on 3 card raises of responder's suit.  Add this to your new partner discussion checklist (for any partner good enough to appreciate the question).


text:
In which, or all of the following situations does a raise of a no trump bid constitute a Blackwood call?
Opener bids one club, responder bids one heart. Opener rebids 2NT (18-19 high card points, balanced). Responder raises to 4NT. I take that to be a "quantitative" bid, aiming for 6NT. The reason is that no suit has yet been agreed upon and 4Nt is actually a raise of 2NT. Am I right?
Opener bids one heart, responder bids 2NT (Jacoby, 13-15, four hearts). A opener's "raise" to 4NT IS Blackwood, because "2NT" was really a "raise" of hearts, not a true no trump bid. Is this correct?
Opener bids one heart, responder answers 4 diamonds (splinter), and opener bids 4NT. This is Blackwood because the splinter bid is (de facto) a raise of the heart suit, right?

If you limit Jacoby responses to 13-15 HCP, how should partner respond to 1H with QJTx KJxx AQJ Kx?

@ruds: The equivalent response would be 3 NT.

So when slam is more likely you like to take up more space in the auction? That seems... bad.

You have correctly identified that the first 4NT is quantitative and the other two are Blackwood. However, I must caution you against falling into the trap mentioned by the Hideous Hog in Victor Mollo's Swings and Arrows:

Don't fall into the common error of assuming that the average player invokes Blackwood to inquire about aces. Aces are purely coincidental. Blackwood is primarily an emotional outlet. The message is: I'm excited. I have a splendid hand and you sound pretty good. Let's go places! Whoopee! Four notrumps!

First of all, Jacoby 2NT is not typically limited to 13-15 HCP. Usually responder is unlimited, and so you should only jump to Blackwood because the number of aces will allow you to place the contract in 5, 6, or 7 hearts. A typical hand would be x AKJxxx KQJxx A. You are safe at the five level; if partner has one ace you will make 6; and if partner has both missing aces, you will make 7. A hand like Axx KQJxx AJx Kx is not suitable for a jump to Blackwood despite its 18 HCP. If partner holds xxx Axxx Qxxx AQ, the five-level is too high. On the other hand, if partner holds Kx Axxxx Kx Axx, 7 is a laydown (assuming opening leader doesn't hold an 8-card side suit allowing an opening ruff). Don't think that asking for kings on the last hand is right, either, since they could hold Kxxx Axxx Kxx Ax, putting you on a finesse for 12 tricks.
On the splinter hand, responder is again unlimited. If you hold x KQJxxx Axxx AK, you can call Blackwood because you have 5-level safety and the missing aces will tell you which level to play at. But with a less distributional hand, the location (and quantity) of partner's filler cards will be important to determine the correct level.

On the splinter hand, something like Kx KQJxxx xxx AK is perhaps slightly more likely as a hand to bid Blackwood.

There are some borderline auctions, and some where different partnerships will go different ways (and sometimes, different partners in the same partnership!)  But standard(1) is:

When opener has bid NT naturally, 4♣ is Gerber, 4NT is quantitative.  Note: this is extended by many to "first and last (natural) NT" - if the first "real" bid was NT (even if there's bidding after) or the last bid was NT, then Gerber/quantitative applies.
When there is a known 8-card fit, 4NT is Blackwood (likely keycard of some sort).
If neither apply, a jump to 4NT is Blackwood (arguments on keycard need to be had with the partnership)
If both apply, you had better have an agreement.  This is where most partnership disagreements show up (and where some players thing "4♣ is Gerber and 4NT is Blackwood."  Find a better solution to this one!)

The ACBL's new Alert Procedures define an "Opening No Trump Sequence"; it's a good way of discussing "first NT".
So yes, as others have mentioned, you have got it right in your three cases.  One, there's no fit, and NT was bid naturally by opener; 4NT is quantitative (and 4♣ would be Gerber, even though opener started with 1♣.  If you have a problem with that, find another solution - common in my area is "the only non-GF response to 2NT is 'pass'."  So your "agree clubs, slam try" call is 3♣)
The other two are not just de facto heart raises, they are by agreement heart raises.  They happen to be heart raises that show something else as well, but that's irrelevant to the "izzet Blackwood" question, over the fact that they promise a heart fit.
As always, your agreements in some odd auctions will depend on your other agreements.  For instance:

1NT-2♦; 2♥-4NT is quantitative with 5 hearts if you play Texas transfers; to RKC in hearts, you go 1NT-4♦; 4♥-4NT.
1NT-2♣ 2♥-? Do you have a call that agrees hearts and is forcing? If so, do that, and after doing that 4NT is Blackwood (don't do it and 4NT is quantitative, probably with 4 spades.  4♣ is Gerber, again, probably with 4 spades).  If you don't, then just assign meanings with partner.
it's worse with 2NT-3♣; 3♥-?  What do you do with a 4=5 black hand with slam-going values?  How do you set hearts, and keycard?  Luckily, this is rare; unluckily, it is common in the rare cases when one partner has half the deck.

"Auctions that start with a natural NT bid" are hard to completely define, if you "always want an ace-ask".  However, as ruds has mentioned, players - especially lower-level players (2) - over-ace-ask, or too-soon ace-ask.  Mostly it's ruds' Mollo quote, my paraphrase of which is: "4NT says 'hey partner, we might have slam!'  And 4♣ is better because we can still get out in 4♠ if partner doesn't agree!"; but a lot of it is just "I don't know how to get the information I need for slam" (or "I don't know the information I need for slam").  You can tell these pairs by the long time it takes to bid 5, or 6, of the trump suit after the "we're missing one ace" Blackwood response - they're concerned they might be off cashing AK.(3)
The other phrase I use for this (which I'm sure I cribbed off one of hundreds of experts) is: "Remember, Blackwood (and Gerber) is not a way to get to good slams.  It's a way to stay out of bad slams" where you can take 14 of the last 11 tricks after they cash both their aces.  As such, to bid it you need to have enough knowledge to commit to bidding 6 if you're off one ace/keycard.  If you're not yet ready to commit, you need to find another way to explore.
All of that is over your specific question of course, which accurately defines the three sequences.
(1) note that I'm a bit of an iconoclast.  I play "Gerber is baby food" (or "No Gerber Ever") if I get a choice.  Costs me maybe one time in two years; just stopping my partners from bidding Gerber and forcing them to actually describe their hand (or the actual, useful meaning for 4♣ in the sequence) has saved me that half-board maybe monthly.  I'm describing standard here.
(2) but a National-level expert I play with occasionally has Gerbered 3 times, and at least twice was wrong to do so (or at least should have done something else first).
(3) this answer is intended as a player answer, but from a directing standpoint, look up "Hesitation Blackwood" for the problem that arises when partner understands the problem and fixes it for the pause-5 bidder.

Ruds has given an excellent answer explaining why you ought not to bid 4NT after an unlimited Jacoby 2NT or a splinter. Nevertheless, should you choose to ignore his advice, I for one would take your 4NT bids exactly as you defined them - quantitative over a no trump sequence, Blackwood if a suit has been agreed.


text:
Are there systems or experts that use a "weighted" resource strategy in Catan?
There is a strategy called (highest) resource strategy in Catan. That is, build your initial settlements on hexes where the sum of the favorable die rolls is a maximum. That assumes 1) that all resources are about equal in value, and 2) an abundant resource is as valuable (or nearly so) as a scarce resource. But both of these assumptions seem counterintuitive.
In this video, a you tuber opined that "one ore > two sheep." That is, he would build his second settlement on a site with less ore than sheep, because the former is more valuable. This person has been blogging and playing for some time, so I assume that he is at least "somewhat" expert. His view also corresponds with my own rough calculations that the relative value of resources is: ore, 5, wheat 4, wood and brick, 3, and sheep 2. Taking the base of 3 for brick wood, I come up with value coefficients of 1.6, 1.3, and 0.7 for ore, wheat, and sheep respectively. I then multiply this vector by the values of ore, wheat and sheep to come up with a "dot product" that gives me an adjusted resource value. In this example, an ore hex with two die rolls is worth more than a sheep hex with four die rolls, because 2 x 1.6 (=3.2) >4x0.7 (=2.8).
Do any systems or experts "re-weight" resources using an algorithm similar to mine and then choose intersections for early settlements accordingly?

The question seems to answer itself. You provide an example of a player you state is an expert ("somewhat") using a weighted scheme. So yes, there is at least one "expert" who uses such a scheme. (Beyond that, say listing all such experts or schemes, is off topic here)

That's a good question, even though I'm not sure you can come up with a universal "weighting" because there are many parameters that determine the relative demand between resources. Such parameters include the layout of your tiles (and the numbers they are assigned) and the stage of the game.
For example, if the overall odds for sheep are really low in a given game, its "weight" vs ore will be different than a game where you have an excess of sheep. Additionally, brick and wood tend to be very valuable at the start of the game when the best strategy is generally to expand, while ore generally becomes more important at later stages.
So I think you could indeed compute relative weights for each resource but these would vary according to many factors. And I think these factors lead players to implicitly choose certain positions vs others positions that may have better odds.

Given the question is based on the selection of initial settlements, discussion of game stage appears to be irrelevant.

@Nij When playing with randomly distributed hex tiles and values, one could incorporate the assumed development into the game in the initial settlement placement, which might deviate from a "general" strategy/weighting. This seems relevant enough to me to the question at hand.

@Nij Given that the objective is still to win the game, optimizing for early game success without taking into consideration middle and late game needs is a recipe for failure. When discussing early game strategy it is always relevant to discuss how  particular strategy affects later stages of play.


text:
Am I allowed to open at the "one" level with hand like AKQxxxx xx xx xx?
I am concerned about a ACBL rules that says that an opening "one" bid cannot be weaker than the top of a "weak two" bid. With this hand, many might open with a "weak two, " and many others might preempt at the three level.
The hand has only nine points according to the Milton Work count but I can get to 12-13 in one of two ways: 1) Counting 1 point for each of the three doubletons brings me to 12. 2) Alternatively, I add two points for each suit card over five, so the sixth and seventh give me four points, bringing the total to 13?
The rules also say that a slightly stronger hand of only 10 hcps, AKQxxxx JTxxx x void is allowed to open with a "strong" two clubs. The reason is that it can make game with "minimal" help from partner, e.g. a low eighth trump and the Q above the JT.
Finally I don't want to weak bid when I have such a good suit and my hand is "average" in high card points (for defensive purposes). AKQ are nine points and I add a tenth for the long sequence, because the J is likely to be swept up by my three top cards.
Am I, in fact, allowed to open this hand with one of a suit?
See the section on allowed bidding agreements.

Nothing in the ACBL chart says "that an opening "one" bid cannot be weaker than the top of a "weak two" bid.

@DavidSiegel: This question arose out of a discussion I had with "Forget" https://boardgames.stackexchange.com/questions/49481/did-i-really-have-too-many-points-for-the-following-bid/49495#49495 He first said that my weak two was "illegal" because it was stronger than the minimum opener. It was the other way around, that an opener could not be weaker than the top of a normal weak two.

@Forget was incorrect in this case (although s/he is usually very accurate). See my recent answer to this Q for detail.

This is too strong for a weak two. Open 1S if your judgement tells you that is the best bid. I doubt there can exist a rule saying that this would mislead the opponents. I don't know about ACBL, but I doubt they would tie the players hands like this. Having said that, I think allowing a 2C-opening with AKQxxx, JTxxx, x,- WITHOUT AN ALERT should be a violation of rules. As that grossly deviates from the 20+hcp and more quick tricks than losers expectation I would have. If an opponent did that to me, and I misjudged, I would call the sheriff.

And, locally, we are quite permissive, but strict about alerts coming from partnership experience. The weakest hand (in terms of hcp) I recall opening with 1S was QJxxxxx, KQxxx,-,x. My partner agreed with my assessment. I wanted to mention both suits, and couldn't think of another way. Switch the majors add 1 point into one of them, and my current partner will introduce both suits in competition if I passed on his opening, reversing!. Having seen that happen, I will, of course, duly alert the bid.

@JyrkiLahtonen: Your hand has eight high card points. But I would add three points for the void, and two for the singleton, which gives you 13. That's a "legit" opening bid, IMHO. And the alert "makes it right."

The ACBL Convention Charts are a list of allowed agreements in the ACBL (but not necessarily other organizations).  You would need to determine which convention chart was being used by the particular competition you were involved in.
Let's check a few.  Please note the following applies to 1H and 1S and not to the minors, which have different rules related to them (though in this case, likely the same - just being clear it's slightly different spots in the rules.)

On the Basic chart, the rule that would control what you may mean by "1 Spade", is the following:

Opening Bids

Any Natural opening bid in a suit, as long as it shows at least Average Strength.

Further, let's define terms:

(1c). “Average Strength”: A hand that has at least 10 HCP or meets the “Rule of 19”.
(17). “High Card Points (HCP)”: The total number of points in a hand based on honors,
counting 4 for an Ace, 3 for a King, 2 for a Queen, and 1 for a Jack.
(27). *** “Rule of N”: A method of determining hand strength computed by adding the High
Card Points of the hand to the number of cards in the two longest suits. To meet the
“Rule of N”, this total must be at least N. *

There is no relationship to the Weak Two range here; what is specifically controlled is whether the "1 Spade" bid is considered permissible.
So we must evaluate your holding here, and see if it fits the range for "Average Hand".

HCP: It has 9 high card points, and does not meet this standard.
Rule of 19:  The distribution is 7-2-2-2, and with 9 HCP, it is evaluated at 9+7+2 = 18.  Thus, it does not meet this standard.

Thus, it is not permissible under the ACBL Basic Convention Chart to have an agreement to open this hand 1S, as a natural bid.

On the Basic+ chart, the rule is:

Opening Bids

Any opening bid in a suit which is Natural, as long as it shows at least Near-Average
Strength.

Defining Near-Average strength:

(1b) “Near Average Strength”: A hand that has at least 8 HCP or meets the “Rule of
17”

This hand is "Near Average Strength", both because it has 9 HCP and 18 "Rule of N" points.  This would be allowed on Basic+ to be opened 1s.

On the Open chart, the following rule applies as to what is disallowed - Open allows all agreements other than what is disallowed.

*** A Natural or Quasi-Natural 1-level opening bid in any seat that could contain less than Near-Average Strength.

This hand is Near-Average Strength, as previously discussed, and so is permitted.
Open+ is more permissive than Open, and so also permits this hand to be opened 1s.

In summary, here are the options:

Chart
1S permitted?

Basic
No

Basic+
Yes

Open
Yes

Open+
Yes

Note that from my understanding, the "Basic" convention chart shouldn't be particularly commonly used in tournaments or even in club play except where you're playing 0-100 games and such; Basic is, well, very basic, and has a pretty limited range of agreements available.  I don't know what's common right now in general, but I'd think most club play is at least Basic+ if not Open.

OK,  under "Basic" I need a distribution of 7-3-2-1 to qualify under "Rule of 19." Would they accept an argument that some experts make that AKQ-7  in sequence are really 10 (because you would capture the J assuming normal 3-2-1 division of cards between the other 3 players)? The other argument is that even though my hand is "only" 9 hcps, I have seven tricks in hand, assuming a Yarborough but at least one spade for partner. So why shouldn't I bid a contract I can probably make?

@TomAu No, on Basic you are limited to the rules as written.  If your convention is to open 7-2-2-2 as 1s, then it is illegal (and that includes if you would commonly open that way, written or not).  The point of "Basic" is to make it somewhat less complicated for beginners and casual players, for the most part, and so it is much more rigid.  Open, which is probably what you're more likely to play under, is not this way, and is closer to what GCC was before (in terms of permitting most things).

@Joe. Tournament events with master-point limits up to 750 may be under the Basic chart.

@DavidSiegel Can be, or is?  I'm not doing much in the way of tournament play right now, but it would surprise me if 0-500 or 0-750 games were commonly Basic, unless it's moved a lot since when I did more tournament play... either way, definitely check what the tournament is played under, and if you do commonly play in Basic chart games, don't get used to opening this 1S.

@Joe. I would have to check recent conditions of contest. The chart says where it may apply. I wouldn't open the hand inn the title as 1S under any Chart, even if it is legal to have an agreement to do so. But some would.

@DavidSiegel Agree there!

@DavidSiegel: Assuming I'm allowed, I would open 1 spade with the hand for these reasons: 1) I can make my bid unless partner "hurts" me with a void in spades. 2) If partner can respond at the "two" level, we can go to game, if not, a part score. 3) I have "normal" defensive values (2 quick tricks). 4) I have the "boss" suit. Opponents have to go one level higher to outbid me. Change my spades with clubs, and I would preempt with three clubs.

@ToimAQu I see your point. That is a question of judgement and partnership agreement. I will add that although some hands could be legally opened at either the 1 or 2 level, a sound system would specify which for most if not all cases.

@TomAu I'd argue that "I can make my bid" is not very relevant - for the most part bidding is about accurately identifying game contracts (most important), slam contracts (slightly less important), and much lower down the line part scores, plus somewhere adjacent to that interfering with opponents' abilities to do same safely.  The main question is, what bid gives you the best chance to identify the 3NT/4S hands; and secondarily, what bid gives you the best chance against opponents when they have a game?

Here, the issue is that 4S is just as achievable after 2S as after 1S; but what's more important is that 1S cannot imply this hand.  2S -> 4S is trivial (2S -> 2NT (Ogust) -> 3NT (AKQxxx) -> pass/4S/6S/etc.).  1S then is able to have a much more well defined range of possible hands.

If you're playing on the Basic Chart (0-"750 or lower" limit game) then you have to be careful.  Otherwise you're fine.  If you're thinking about opening much lighter than standard in the first two seats, the under-750 games aren't really for you.
As others have said, there's nothing regulatory about top of "weak 2" being higher than lower of "1".  There are hands that could be opened 2, or 1, depending on state of play, opponents, vulnerability, differing styles between the partners, or what side of the bed one got out of today.  And that's fine.
One thing I would warn you of is Walrus thinking.  AKQxxxx Txx 9x x is not a 9-count even the way KQJxxxx Kxx 9x x is (to me that's an obvious 3S opener). X-counts are not alike, and some get upgraded and some don't.
The only "too strong for" thing the Charts talk about is "if two hands are exactly the same except that an honour is replaced with a low card in the same suit, you cannot open the hand with the low card and pass the hand with an honour."  This stops all the complaining that "it's ludicrous that I can't open AKxxx QT9xxx x x if I pass QJx QJx QJxx QJx".  But that's only for the Open charts, related to passes rather than weak 2s, and not relevant to the question.
To discuss the second half of your post: yes, the Open chart says that you can agree to open an artificial 2C on any hand, provided it promises Average Strength.  Yes, that means that you can decide to open xx AKQJxxxxx xx -, 2C.  This is a common agreement, usually by weaker players, that 2C doesn't promise defence.  It causes two problems, one that it goes 2C-(2S)-p-(4S) and neither player knows if 4S will make or go -1100; and when the star player opponent finds out that they opened 2C and he can make 6H, a meltdown of catastrophic proportions occurs.  It's a bad agreement, but it's legal.
The problem has always been disclosure - how do the "strong, doesn't promise defence" people make their "obvious 2C" agreement known to the opponents?
Now, there's two solutions:

2C Artificial, not Very Strong (which requires AAK or AKKK minimum for it's distributional versions) is not allowed on the Basic or Basic+ chart.  I'm guessing the regulators hope that if newer players can't play this poor agreement, they won't.
for Open(+), the new Alert Procedure now explicitly states that if your 2C opener isn't guaranteed Very Strong, you must Alert it.  Every time.  And explain that it may not have defence.  Now, the strong opponents are on an equal footing.

If I bid 2c with AKQxxxx JTxxx x void, I'm saying that I can make game in my rebid suit (spades) if partner has as little as one spade and Qxx of hearts. I'm saying nothing about defense. (Even so, with so many spades and hearts, there is no way that the opponents can bid game in a major.) One reason the ACBL allows this 2 club bid is because I have more than five "controls" (2 for the void, 1 for the singleton, 2 for the ace and 1 for the king). And when I bid 1 spade with the other hand, I can make it, barring freak distribution, even though my AKQ are technically "below average" in hcps.

P.S.  AKQxxxx xxxxx x void,is considered "very strong" (despite its few hcps), because of all the controls (6 vs a minimum of 5) alluded to in the foregoing.

Control Count of 3.  A=2, K=1.  You have to read the definitions carefully.  That restriction is in there explicitly to exclude hands without defence.  You are allowed to have whatever agreement you wish, provided it's legal in the game you play.  But if you're in the ACBL, and you agree that your first hand is a 2C opener, you have to Alert your 2C openers - all of them, even the flat 22s.  If you don't, and the opponents have game or slam on and you "preempted them" out of it, it will be adjusted.  If you have that agreement in a Basic or Basic+ game, it will be adjusted for two reasons.

Taking my director hat off, it's a bad agreement, as I mentioned in my answer.  It's an agreement weaker players have, because their weaker opponents treat a 2C opener as shutout.  It loses when the stronger players compete - hard - and your side doubles for -590 into -420 or pulls the double for -100 into +100.

Where do you stand if I do the control count "your" way, 2 As and 1K, 11 hcps, 5 controls, all from high cards?

As a player, my agreement for strong 2C is "22+ or 8.5 Playing tricks with defence.  Defence means 'if they bid game, and partner doubles with a flat zero-count, it's going down barring duelling voids or the like."  As a director, my only concerns are that you're playing something legal, and that the opponents are aware of what you're playing.  Whether what you play is "good bridge" in my, or anyone else's, opinion, doesn't matter.

Nothing in the ACBL chart of allowed conventions says "that an opening 'one' bid cannot be weaker than the top of a 'weak two' bid. Under the Basic chart, any natural bid showing at least average strength (defiend as 10 HCP or rule of 19, slightly stronger than the specified hand) is permitted. The rule for a weak two in the Basic chart is:

Any Weak Natural opening bid in a suit at the 2-level showing at least 4 HCP and has a Range not greater than 7 HCP.

This would clearly include a hand with 10 HCP, as a range of 7 with its lower limit at 4 has its upper limit at 10.  But it would seem that a range of 5-11 would also be permitted under this rule.
Thus a hand of
AKQxxxx Jx xx xx
would be legal as either a 1S or a 2S opening under the ACBL Basic chart rules. A hand of
AKQxxxx xx xx xx
would be legal as either a 1S or a 2S opening under the ACBL Basic+ chart rules.
In general a natural Hand including 10 HCP will be legal at either the 1 or 2 level in ACBL Basic (and all higher charts), and one with 9 HCP will be legal at   either the 1 or 2 level under ACBL Basic+ rules (or any higher chart).
Any statement that a weak two bid must be weaker than any opening bid under ACBL rules is incorrect. It might be advisable for a pair to agree to such a rule as part of their system, but it is not required. In fact there is no length requirement for a weak 2, although most systems include such a requirement.
A hand of xx KQxxx KQx xxx is valid as either a 1H or a 2H bid under Basic+. (I personally would pass this except in third seat, however.)
If an event is being played subject to the ACBL Basic conventions chart (and not any higher chart) then an agreement to open AKQT987 T9 T9 T9 with a bid of  1S is illegal.
The definitions of HCP in the charts is strict, with no room for interpretation. It refers to the classic Work-count of 4, 3, 2, 1 for A, J, Q, J with no adjustments.
Under the Basic chart players may not agree to make a 1-level bid unless it contains 10 HCP, or passes the rule of 19 (HCP plus length of two longest suits is at least 19).  This hand does neither. The chart says:

If an Agreement would be disallowed unless it satisfies a specific High Card Point or shape requirement, a player may not use judgment to include hands with fewer High Card Points or a different shape.

Would a player be penalized for regarding e.g. AKQT987 T9 T9 T9 as being worth an "extra" point compared to AKQ5432 32 32 32 since the former would be able to run the spades in many more situations than the latter, and the tens would have some likelihood of interfering with opponents' finesses?

@Super If an event is subject to the ACBL Basic conventions chart (not any higher chart) then an agreement to open AKQT987 T9 T9 T9 at 1S is illegal. The definitions of HCP in the charts is strict, with no room for interpretation. Under the Basic chart players may not agree to make a 1-level bid unless it contains 10HCP, or passes the rule of 19. This hand does neither. The chart says "If an Agreement would be disallowed unless it satisfies a specific High Card Point or shape requirement, a player may not use judgment to include hands with fewer High Card Points or a different shape."

That hand counts 21 playing points so is short of a two demand and  must be opened at the one level.
A better question is whether one club would be reasonable to open with then jump shift into the strong spade suit.

This is referring to the "requirements"  of a particular bidding system, not what ACBL rules require. Opening this 1C with a plan to JS to 3S would in my view be bad bridge, unless playing some form of a strong club system.But it would be legal.What bidding system does this answer assume?

I played American Standard with my son before 5 card majors came into vogue.  And I saw many other pairs open 1S with that hand in their assorted systems.

That was a long time ago. 5 card majors were already fairly common, although recent, when i started playing duplicate in 1980. The rules on permitted bids were quite different then, and not as well enforced. Even now the rules are not always enforced, esp at the club level.

@DavidSiegel: To paraphrase: 5 card majors were already fairly common, although recent, when i started playing duplicate in 1972. Even players who started playing competitively in the late 1960's were aware of K-S and Walsh variants of 5-card Majors, on the East Coast and West Coast respectively, by then. Lawrence and Goldman played an early 2/1 Game Forcing system from at least 1968, playing on the Dallas Aces.


text:
Should my partner have initially kept silent with the following "2NT" hand?
Neither side vulnerable, sitting in the second seat, my partner opened one club with the following: (s) 9 (h) 9 (d)A7543 (c)AJT987.
Third seat overcalled, two hearts (showing 6). I passed in fourth seat with a 4-4-3-2 shape and 10 high card points in the red suits. First seat also passed.
At this point, partner passed, not able to show her diamond suit, and our opponents got to their best contract (bid two hearts made three) and a top on the board.
But if partner had passed on her first opportunity to speak, couldn't she now bid 2NT (unusual)? I had only three diamonds, with which to bid three diamonds, but they were KQJ. Other pairs with our cards bid and made 4 diamonds.
Is it fair to so that partner's "premature" opening bid precluded her from making the "right" bid later? Or are there advantages of opening 1 club in her position that I have overlooked? (People in my club often open with 9-10 high card points if they have 5-5 or better in length.)

I can't imagine passing with your 10 count.  What was it?  Different partnerships have different styles on opening with 9 or 10 hcp shapely hands, and all of them can work.

@AlexanderWoo: I can't either, but I now recall that third seat (my RHO) jumped to two hearts.

Are you sure it would've gotten back to partner below 2N?

@AlexanderWoo: The final contract was two hearts (made three).

Considering that I considered opening that hand 3C, there is no way I would pass if it had gone 1C-2H-P-P back to me.  (But with 10 hcp, I think you should still make a negative double of 2H if you have 4 spades.)

My $0.02 is that yes, you should pass that hand. In fact, with even stronger hands:

I would definitely pass ♠x ♥x ♦AQxxx ♣Axxxxx
I would strongly consider passing ♠x ♥x ♦Axxxx ♣AQxxxx

I find a 3♣ preempt opening an acceptable option (too aggressive for my taste even NV against V, and it sacrifices a potential diamond fit, but at least it reasonably describes the hand). I do not think standard systems have an opening bid for "weak two-suiter" that includes both minors.
I definitely think opening "strong" with that hand is a mistake, unless you have a system that allows very light openings, in which case it becomes a question of "what does our system say" rather than "what should have been done with a standard system". (My system, and the one of 99%+ of players where I practice, is that you need 12 HCP + length to open, or 11HCP "and a reason".)
The main problem I see with opening your 9HCP count is that partner will push you into 3NT around 30% of the time (1), expecting you to have 11HCP, and that contract is going to be hell. Only specific hands by partner makes it: you need a stopper in both majors, and to make enough tricks with that, so the bare minimum seems to be ♠A ♥A ♣K in a 3-club hand. Most often you will go down, possibly by a lot.
(1) Rough estimation, but probably not too far off. Partner will put you in NT as soon as they have 12-18HCP with both majors stoppers and a somewhat balanced hand (2). Given your 9HCP count, "12-18HCP" is somewhere around 45%; given your minor aces, "partner stops both majors" is fairly frequent if they have those points; "somewhat balanced" (2) happens fairly often.
(2) Meaning, they do not have 4 clubs or 5 diamonds. OK, they might have a 6-card major, but 4♥ or 4♠ in 6-1 fit with 25-ish HCP is not an improvement over 3NT.
Furthermore, even if partner has 8-12HCP and you are fighting for a part score, does it really help to get in at this stage?

If the opponents start a "constructive" auction (say, 1♥/♠ followed by 1NT, 2♦ or 2♥/♠), firstly partner might speak. If they do not, you may still intervene. This depends on your system, but coming in after an initial pass definitely describes your hand better to partner (shapely, low-point hand) than whatever convention you can cook up that pushes you to open 1♣ with both a 9HCP count and a 18HCP count.
If LHO pre-empts partner with 2♥ or 2♠, partner might still speak. They have a good idea of where points are (you and RHO passed so are both 11HCP or lower, LHO is pre-empting so 6-10, they can see their hand). They will choose depending on their shape, their expectations to set the pre-empt, and so on. They might go wrong, but they definitely have more information than you to choose.
Finally, if opponents find an "attack-defense" auction of pass-pass-1♥-pass-4♥ (or 2♥-4♥, or the same in spades): the question is not the contract (opponents would have found it anyway), but whether you should double. I would argue that you should not: if partner had 2 or fewer cards in the named color, and some points (10-ish?), surely they would have spoken up (by doubling for instance). Conversely, if partner did speak up (e.g. with pass-pass-1♥-X-4♥), you can double (arguable, but with 18-ish points in my line and my honors being aces, I would do it).

Even if you open, which gets you in... your rebid becomes very hard. The other comments discuss some approaches - I have no qualms with them, but frankly I think there’s no hope to fix the wrong opening bid, there’s going to be a large fraction of hands that end up in disaster. Again, the basic problem is that 1♣ is already a large range of points and shapes; if you expand it to include hands such as this one, you need very precise rebids to deal with the initial uncertainty.
The only good cases for opening strong seem to be:

general pass when partner has 10-12HCP and a minor fit better than the opponent’s (likely) major fit, so that you would make a minor contract (or push the opponents to a failing major contract)
partner has an incredibly fitted and shaped hand, say ♠Axx ♥x ♦KQxx ♣Kxxxx; slam is within the range of possibilities, but third-seat opponent has 7+ in either major and will preempt so hard that finding game in your line is already difficult.
attack-defense auction, and partner has enough points to set the contract but not enough points+shape to speak up

First case is maybe 5% of cases, costs very little (depends on scoring, general strength of the field etc.), and still requires you to make delicate auction calls after the initial opening (3♣/♦X-2 is a real possibility if you and partner are not very careful). Second and third cases do cost more, but they are very rare - less than 1% of cases combined? Those numbers may be off, but the point is that they are much smaller than the double-digit percentage of bad 3NT games that you push partner into.

You forget the very common case at matchpoints where you going down in a minor suit contract is better than letting them make a major suit contract.  Neither vulnerable, so 3C down 2 is better than letting them make 2 of a major.  Or you have some fit and going down 2 doubled in 5 of a minor is better than letting them make 4 of a major.

I would open both of those hands 1D. Yes, the hand may (but need not) be a disappointing dummy to partner playing 3NT. Or any hand that is a misfit. These type of hands want to find a fit. If there is none, it is ok to end up too high, because you will have some field protection. Of course, if you desperately need a swing (or a top), then passing has tactical merit.

With 56 in the minors (or majors!) and not enough strength to reverse, it is often a good idea to open with the higher ranking suit regardless of which suit is longer.
So your partner might have opened with 1D instead. The benefit of that choice became clear. The way the auction went they would have been able to reopen with 3C, and you could have made a preference 3D, likely ending the auction.
It won't necessarily be the opponents making it awkward to bid both your suits. Even a very routine early auction of 1C(partner) - 1S(you) may make your partner squirm, knowing that your side might belong in diamonds, but now forced to rebid 2C lest they distort the strength of the hand.
Of course, you could have made a negative double. This time you would have undoubtedly been more comfortable with that action after a 1D opening. After all, you are fine with playing in a diamond contract if partner has no fit to your spades.

This treatment is quite common. Of course, there is the risk that you end up playing in a 53 (or even 52) fit instead of a 63 fit, because partner will assume the first bid suit to be at least as long as the other. You can't have everything, but with two 5+ card suits you often want to mention them both.
It is, for obvious reasons, a better known tactic to open 1S with a minimum opening hand and 56 in the majors. But the same thinking (preparing for a rebid) can work here as well. But, never do this with 45 suits (unless systemically agreed upon with your partner), partner must be able to rely on the first suit having at least five cards.

I think if I decide to open that hand, I have committed to "6-5 come alive".  I can't pass this.
What to do?  3♦ is overbidding, sure, but it's in the mix; 3♣ is okay, but doesn't show the suits; but I think I double.  If partner has a penalty double hand, my two aces will be happy for them, and it might start spade, spade ruff opposite a weak (I assume?) 2♥!  If partner bids 2♠, I'll bid 3♣ now and hope to survive.
Sure, if you pass this hand, you have a great 2NT call over a  1 bid; but on this hand you won't get one, you'll get a third seat weak 2.  Do you really want to say 2NT over p-p-2♥-p; p?  What about p-p-2♥-p; 3♥?  At least opening gets your foot in the door.
I'd love to see your hand - "4=4=3=2 10 count in the reds" could mean a lot of things.  But you likely have a nice (if very minimum) double yourself; are you going to be happy if partner passes this out with 4=2=2=5 or 4=2=3=4 12 or 13?  Or do you expect partner to protect-double with those hands?  Your best contract, really, could be 2♠ with partner's 3=1=4=5 (if they don't guess the moysian and lead trump, at least!)


text:
How strong a suit do you need for an overcall above the one level in direct seat?
If someone opens say, one club, the next person will mostly likely overcall, say, one spade with a decent five card suit and about ten high card points, even though the hand is slightly too weak to open.
But if an opponent opened one spade, and I needed to overcall  diamonds at the two level, my personal standard is six diamonds, headed at least one sure "top' trick. That is Axxxxx, KQxxxx, or even QJTxxx. For a pre-empt or a three level overcall I would need a seventh diamond. I would also overcall with five and six respectively, with a strong top sequence such as KQJxx(x) ore better.
But in a recent game, we were not vulnerable against vulnerable opponents at matchpoints, Left hand opponent dealt and opened two spades (weak). My new partner* overcalled three diamonds with the following: ♠A2  ♡ Qt9 ♢ KT873 ♣ k74.
Right hand opponent raised to four spades, and I decided to sacrifice at five diamonds, with the following: ♠T65  ♡ J8 ♢ A964 ♣ T953. I have been told that my overcalling standards are "light," but I still expected more than partner had. If the suit were King high, I would expect something like KT87532. With a five card suit, I would expect the holding to be KQJT2 or better.
My goal was for us to go down no more than three, -500, against the opponents' 620 spade game (which made at other tables).
We went down five with his actual holding. Part of this was unlucky, my right hand opponent had QJ52 of diamonds versus a void in the other hand, meaning that we would have been down four against a 3-1 distribution, (and down three against 2-2). Counting this, we went down one more than expected.
Did it make sense for me to expect partner to have another honor or two more small cards for his overcall than he did, to limit the likely loss to down three? Or was the problem with my raise, egen given my goal of down three. One of my opponents offered the opinion that my trump and high card holdings were fine but that I should have had more ruffing value for my raise.
*(I'm on "rotation" at a new club)

How do opponents come to ten tricks in a 4 spade contract against the defense of spade ace followed by a low one, creating two diamond losers for them? That defense would seem called for from partner after the pre-emptive raise to four on his left.

@ForgetIwaseverhere: They made five trump tricks, three heart tricks (A, K, 4th, losing the third), and two out of three club honor tricks Declarer was void in diamonds,, hence no losers, while dummy had QJ52, which hurt our 5 diamond declaration. My claim is that partner should have had at least two of the top 3 honors for his overcall with 5, so that dummy would have Jxxx at most. Or with K-7th and my A-4th, we drop their QJ, and hold our loss to three, which would have given us a top score.. . –
Tom Au

Ah! I thought the diamond void was in Dummy rather than Declarer. Thank you for the clarification.

@ForgetIwaseverhere: The new contnbutor (below) pointed out that I should not have put partner on K-7th of diamonds initially. But after RHO "advertised" a four card spade suit with her 4S bid, it seemed like partner had a void in spades. Then his shape would be 0-3-7-3, or 0-4-6-3. In the first case, we drop the opposing diamond honors,and in the second case,  78% of the time with QJX, and hold our loss to down three. . With 0-4-5-4, he would presumably double rather than bid three diamonds.

@TomAu You are overthinking this. If it was a normal auction, then yes, you should think hard about what partner might have and what to do (for instance, what to bid after (2♠) 3♢ (3NT), and if pass what to lead against 3NT, is a really interesting question). But this is not a normal auction and you know it (see my answer). Let the opponents handle the trainwreck, especially since your cards do not call for heroism (no distributional value). Sometimes you get swindled (here the opponents are lucky that 4♠ makes, and very lucky that 3NT is not a better contract) but on average it is better.

Partner’s bid with ♠Ax ♡QT9 ♢Kxxxx ♣Kxx is OK
In my book, (2♠) 3♢ shows 11+H, 3 or fewer hearts, and 5 good diamonds. KTxxx is not really "good diamonds", but I would not blame partner - the favorable vulnerability protects them against a 18+H LHO looking for a penalty, and the only alternative is doubling which is not very informative.
The risk of 3♢ is to make -1 in a part score when 3♠ makes -1, or to play 5♢-1 instead of 3NT= thanks to the spade stopper; it sucks, but it is not very likely.
On the other hand, double can easily lead to disaster. As soon as partner holds four hearts, it is going to be very hard to avoid playing in hearts. Yes, double does not show four hearts in my system, but you have no real option but to pass after

(2♠) X (3♠) 4♡
(2♠) X (pass) 3♡

...and either sequence is really likely.
Your turn: what is going on?
The first step is to realize that something is off from the bidding.
The opponents claimed ten spades between them (2♠ shows six spades, 4♠ shows four). If that is correct, partner is void, and the opponents’ spade honors are wasted points.
But... what does partner then have, to say 3♢? Surely, with a 0463 or 0364 hand of 12+H, they would double first; if they are weak but distributed, say with a 6-5 bicolor, they either do your system’s bicolor bid or double. But maybe partner is weak in honors, and possibly longer in diamonds - maybe they have something like ♠- ♡xxx ♢KQJxxxx ♣xxx, give or take a queen.
But then... what is RHO doing? You have nothing in hearts and clubs, so they must have ♠Qxxx ♡AKQx ♢x ♣AKQx, give or take a queen, and they do not even try to investigate a slam opposite a spade pre-empt?
One of the following is happening:

partner failed to double with a strong-ish hand, or they made a big psych;
LHO or RHO lied about their spade holding, and they are in significant danger of going down (the actual game);
RHO has a strong hand, and is actively preventing their team to find a slam.

We assume case #1 to never happen, otherwise your partnership becomes a game of second-guessing.
The proper action in case #3 is definitely to pass; any overcall allows RHO to wake up and try to fix their underbid. (LHO will surely pass, since RHO’s 4♠ could come from a hand with 0H and four spades, and 4♠ will make between +1 and +3.)
I submit to you that while 5♢ might be the correct call in some subset of case #2, it usually is not. The correct call is to pass.
The actual hand: the opponents should have found hearts
I infer from the comments that the spade-player hands were (Alice) ♠KQxxxx ♡xxxx ♢- ♣xxx and (Bob) ♠Jx ♡AKxx ♢QJ9x ♣AQJ.
Bob’s bid leads their pair to play 4♠. It turns out that it makes as long as hearts split 3-2 (or the defense fails to find the ruff, or the club finesse works), but that is a rather lucky placement of cards: in Bob’s hand, put a jack in hearts instead of that of club or spades, or move a small heart to spades (which would usually be an improvement!), and it becomes difficult. Again, in your seat, it is absolutely not clear that 4♠ makes.
I believe the "book" bid for Bob would have been 3NT (18H, a good diamond stopper, regular hand). Now, Bob can anticipate that entry to dummy will be difficult if the ♠A or an offside K♠ is missing; depending on what the hearts are, 3NT might go down against a competent defense. (At any rate, maybe Alice will overcall 4♠ with her void.)
If you do not like 3NT, then Alice and Bob are likely to find 4♡ (absent a jump to 5♢ from OP, which would be more defensible now that we know the opponents have a double fit). This is definitely a better contract: playing a 4-4 fit is much better for ruffing diamonds than a 6-2 fit. The declarer has no real trouble unless the spades split 4-1 and the defender short in spades is also short in hearts.

There were four tables, and three other pairs playing the oppoents' hands. One out of the three pairs found hearts.

@TomAu Thinking again, I am not sure how finding hearts work. After something like (2♠) 3♢ (X) (double would be my second-choice bid for Bob after NT), I would definitely jump to 5♢ with your hand at this vulnerability - Alice cannot really speak again unless double promises four hearts (does it?). Maybe 2♠ (3♢) 3♡ (5♢) is what happened, and then Alice has an easy bid of 5♡.

Let's suppose partner held the same hand, and you held instead xxxx AKx AQx Jxx.  This is a good hand in that situation, but hardly exceptional.
If partner had passed, supposing RHO passes, are you coming in (with what?), or are you missing a cold 3N?
If partner doubles and RHO passes, how do you get to 3N?
(With partner's 3D bid, if RHO passes, you bid 3S with my example hand, and partner bids 3N.)
Over a preempt, overcalls have to be based more on general strength, and less on suit quality.  That's because you have to worry about missing game if you pass a strong hand with a not so good suit, whereas when the opponents open at the 1 level you're much less likely to have a game.
I wouldn't bid 2D with your partner's hand over 1S; I'd double instead.  Over 2S, you don't have as much room to find the right strain after a double (particularly if you're not playing Lebensohl) and 3D is better.
(I absolutely wouldn't bid 5D with your hand there.  Partner's overcall promised an opening hand or better, and they could easily have 3 defensive tricks in their own hand, in which case your sacrifice would be a phantom.)

First, good thoughts about "playing the hand during the auction".  I can understand your issue with "partner needs a good suit and a decent hand to come in at the 3 level", and with that, you can "see" 10 tricks.  But preempts work, and sometimes you just have to get in with imperfect hands, or you get talked out of way too much.
I, too, would double with partner's hand.  I would consider this a dead minimum, and yes, we might be playing 3 or 4 hearts on the Moysian, but I don't want to "show" diamonds only with support for three suits.
Note: I am known to be very conservative against opponents' preempts - I am willing to just let them play a lot to avoid partner having a bad "can we make game" guess.  I also count on partner to balance knowing this.  That does, and will colour my thinking here.
I wouldn't be bidding 5♦ for any reason that involves "going down".  RHO's 4♠ says one of two things (yes, even at unfavourable):

we have 10 trump, and I think "putting them to the last guess" is our best hope for a good score; or
we have 10 tricks if you have the suit you promised.  Hopefully.

How strong does partner have to be to "double and bid"?  20?  How good a suit does partner have to have to "double and bid"? Especially against 2♠-X-3♠-p; p? Do you play lebensohl over weak 2 doubled, so that we might survive a 3 count?  Those questions will determine what hands you have to think about partner holding.
Given answers to all that, if the opponents have the first type of raise, you need to be making 11 tricks for your plus - even with the "known" spade void, you're providing exactly one trick and preventing exactly zero losers (okay, maybe a slow trump loser).  If partner has that hand, they'll do something.  And if we can set 4♠, which is very possible, you've just converted a plus (even +100) into a minus.
If the opponents have the second type of raise, well, maybe opener fudged their first-seat red/white preempt a bit.  Maybe the "outside K" is worthless given your partner has all the cards.  Maybe there's a transportation problem, or.  Maybe, at matchpoints, even the fact that the diamond ace is in the wrong spot might make them misplay and give away the overtrick, and at least we beat the -650s.
Or maybe 5♦ is a good sacrifice, going for -500 or even -300.  The other question to ask is "how many are getting to game their way?" or, if it's a tough hand, "how many are making it"?  Because -300 loses to those pairs just as much as -620 or -650.  And if you expect it to go -500, then a bad break is -800.
And remember, when your side is playing, the points are still "over" the strong hand.  Even if your partner had the KT-sixth or KQJ-fifth you're asking for, you're still losing a spade (only one? pretty nice), and all your round cards (okay, only three clubs). Even that is 3 clubs, two hearts and a spade for -800.  Okay, sure, opener should have an outside card, and it pretty much has to be an ace.  So -500; but now you're taking two Aces and the offside K, and if it's the 5-card suit and the diamonds break 2-2 as well, or if you get to resolve the heart ruff (diamond to the Ace, heart through when you see weakness in dummy; when partner takes their trump and cashes the heart K...) it's -500 into +100.
So, one good case, that does not guarantee to swing the full amount.  Three bad cases (I don't think 5♦ is ever making), one of which might even convert a good score (rather than an average one) to a minus. My "last guess" (which isn't; partner might double or bid 4NT or...but likely it is) is pass.