Where does the phrase "Botvinnik moment" come from? In his analysis of the first game in the 2018 world championship, Antonio Radić says that on move 9. ... b7-b6, Carlsen took six minutes to make a move that he probably already knew: 


  And now Carlsen again goes b6, but it did take him some six minutes for this move. As the other moves Carlsen played relatively quickly, so I don't think this is a new position for him. I think he actually came very much prepared and he was just like a Botvinnik moment, where he takes six minutes to think about a move he already prepared.


The only other instance of "Botvinnik moment" was in a comment on the game Raymond Keene vs Mikhail Botvinnik, Hastings, 1966-1967. The annotation for the last move says, "At which point Botvinnik gasped, raised his hand to his forehead, and resigned," and a commenter added, "We have all been there and had this Botvinnik moment."

This is clearly not what Agadmator is referring to. Was Botvinnik really known for taking a relatively long time for moves that he had prepared? Is "Botvinnik moment" a common phrase for this?
It is/was a common advice, in a game with a classical time-control, to take a small break during the opening instead of flashing out the moves, even if you know them by heart. The idea is to 'warm-up' your brain, to refresh your knowledge about the main ideas and plans in the variation, and to avoid starting out with a very long think after the first move by your opponent that you had not anticipated: first that would give him information about how deep your preparation was, second your very first thought would be about a difficult and important decision - when any mistake might be very consequential.

When he took a few minutes before playing 9...b6 (even though he already knew he would play that move), Carlsen was most probably mentally rehearsing the possible variations from that point, refreshing his positionnal guidelines in the given pawn structure and remembering which previous games of Caruana had already been played from similar positions.

The advice was originally spread within the infamous post-WWII 'Soviet Chess School', so it doesn't come as a surprise if it is somehow associated with Botvinnik - possibly Botvinnik formulated it first or endorsed it in a publication.

I consider the Keene-Botvinnik 'moment' to be something totally different: the commentator only stress out that every player has already found himself in that embarassing situation of having to resign a long-fought game after an oversight.