In the Delayed Open Catalan, after 1. Nf3 d5 2. d4 Nf6 3. c4 e6 4. g3 Be7 5. Bg2 O-O 6. O-O dxc4 7. Ne5 c5 8. dxc5 Bxc5, which move should I play? In the Delayed Open Catalan, after 1. Nf3 d5 2. d4 Nf6 3. c4 e6 4. g3 Be7 5. Bg2 O-O 6. O-O dxc4 7. Ne5 c5 8. dxc5 Bxc5, we reach this position:

[Title "White to move"]
[fen "rnbq1rk1/pp3ppp/4pn3/2b1N3/2p5/6P1/PP2PPBP/RNBQ1RK1 w - - 0 1"]


I made 4 of the 5 strongest engines analyze the position for more than 30 minutes each. Here are the results:









It seems all the engines think that 9. Qxd8 (followed by 9... Rxd8 10. Nc3) is the strongest move. And they all also seem to think that 9. Nc3 is the second best move.

But on the book Wojo's Weapons Volume 1, the authors apparently recommend 9. Nc3, although they don't talk a lot about that move (Wojo actually never faced 7... c5, all his opponents answered 7... Nc6 instead).

So which move do you recommend me to play? What are all the advantages and disadvantages of 9. Nc3 and 9. Qxd8 ? Should I put my faith on the engines or on the authors of Wojo's Weapons?
It is a well deserved doubt that you have in this position. As a Catalan player, I like both continuations (i.e. Qxd8 or Nc3) for well defined reasons. First one important note on Catalan openings or similar positions: 

White's play evolves around the positional play of minor pieces.

Meaning in any position resulting from a Catalan, you should reason by continuously evaluating your minor pieces development progress versus that of your opponent (less focus on pawn structure as the center is already simplified.). Added to which there's white extremely solid and flexible pawn structure, which means any early end-game transitions should be favored for white. 

Now regarding your engine suggestions vs Wojo's version: 

The engine of course looks for maximum edge in any position, in this particular one, the immediate endgame transition right after a queen exchange is favored for white, because of the slight edge in development, and there are no tempos to be gained by black, hence the slight + score you see in all evaluations after Qxd8. To showcase one of them: 

[Title "Qxd8 example line"]
[StartFlipped "0"]
[fen "rnbq1rk1/pp3ppp/4pn2/2b1N3/2p5/6P1/PP2PPBP/RNBQ1RK1 w - - 0 1"]

1.Qxd8 Rxd8 2.Nc3 Be7 3.Bf4 Nd5 4.Rad1 g5 5.Bc1 


It goes without saying that white has the more comfortable position, with all pieces actively in play, better control on d-file, c4, g5 under pressure, and no pawn structure worries for white, whereas black needs to find the right tempos to activate the 3 remaining pieces and all the time cover the weaknesses.

[Title "Qxd8 2nd example"]
[StartFlipped "0"]
[fen "rnbq1rk1/pp3ppp/4pn2/2b1N3/2p5/6P1/PP2PPBP/RNBQ1RK1 w - - 0 1"]

1.Qxd8 Rxd8 2.Nc3 Nbd7 3.Rd1 Be7 4.Nxc4


One example showing black can't just easily activate the pieces after the queen-exchange, after Nxc4, Bg5 coming, constant pressure on b7, better control on d-file...all shows how better (or just more comfortable white is). 

Conclusion for Qxd8: If you're comfortable playing this type of middle-game endgame positions, you should definitely try playing the Qxd8 line, a good experience in these lines will greatly improve your minor pieces positional play (finding right squares, covering the right ones, keeping pressure, coordination, etc.). Note that these are not necessarily easy to play positions if you don't have experience in such positions. Often one gets confused where to place the pieces (knight on c4, or e5, or e3 etc... or maneuvers like Nb5 to Nd4 sometimes very handy, point being white can also easily mess up.)

Now regarding Wojo's suggestion: there's a simple reason behind: white intends to keep the queens on, and play for the slight activity edge, without forcing an endgame right off. After Nc3 there are many different lines that can be played, I will show two of them. In the calm ones, white's plan relies on black cramped white bishop, caged rook on a8, and the rather exposed queen of black after Qc7 or  Qb6 etc. Here you have to be good at keeping your slight edge and slow by slow poking at black's weaknesses, and most importantly finding the right time to exchange and simplify the position to leverage the advantage in an endgame. 

Before I show the lines, note that after Nc3 black cannot just exchange the queens, because the resulting position after Rxd1 is just a much better version of what white had after Qxd8, since the d-file is completely in white's hands. 

[Title "Nc3 example line"]
[StartFlipped "0"]
[fen "rnbq1rk1/pp3ppp/4pn2/2b1N3/2p5/6P1/PP2PPBP/RNBQ1RK1 w - - 0 1"]

1.Nc3 Qc7 2.Nxc4 Nc6 3.Bg5 Be7 4.Nb5 Qb8 5.Qc2


With Rfd1, Rac1 and Nd4 coming up, black is under great positional pressure and will have a very hard time to finish development for a while.

Now one fancy line, where probably white has a great attack with very positive chances (doesn't necessarily mean black is hopeless).

[Title "Nc3 fancy continuation"]
[StartFlipped "0"]
[fen "rnbq1rk1/pp3ppp/4pn2/2b1N3/2p5/6P1/PP2PPBP/RNBQ1RK1 w - - 0 1"]

1.Nc3 Qc7 2.Nxc4 Bxf2+ 3.Rxf2 Qxc4 4.Rxf6 gxf6 5.Bh6 Nd7 6.Bxf8 Kxf8 7.Qd2


Where again white has the initiative with better chances, just look at the activity of each piece...(and the rook coming on f or d-file). This is a fancy and rather complicated line which you can avoid of course if you don't like.

So you see after both Qxd8 and Nc3, it's just a relatively equal game where white has the comfortable play, with easy to find squares for all his pieces, whereas black not necessarily lost, has to keep struggling to finish development without ending up with less pawns eventually. I personally like Nc3 because I usually play these lines with the queens on, and for you, you just have to see which type of lines you prefer to play (or better to say : more comfortable in), this should be your main reasoning when in doubts like "what opening variations to play". 

By no means an eval of +0.50 or +0.20 by the engine means winning for white, no one is able to keep such slight edge from the opening, not even the engines, you see these evaluations because they suggest who has the easier play, the initiative and the quicker development, but chess is way too dynamic to allow for such slight edges to suffice in winning a position. I certainly recommend the Catalan opening, and you should try all these lines for yourself at least a couple of times, so that all the ideas discussed above become clear to you.

Let me know if you have any further questions, and if you want to discuss a specific line.

EDIT: I would certainly recommend studying some of the games of Kramnik, Ivanchuk and Carlsen, all playing the white side of this opening. They will be very resourceful to you.