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The crux of the question is: When does the out occur - when it's called or when the event being called happens.
Similar to a timing play, right...
So extend your logic to this scenario:
0-2 count, 1 out. R1 on 3rd, R2 on 3rd. Batter checks, R2 is stealing. Catcher throws down to 2nd, R2 is caught in a rundown, R1 takes off and scores well before R2 is out. NOW, F2 asks for an appeal and BU (right or wrong) rings up the batter.
So when, now, did the strikeout occur? Right then when it was appealed? If so, no run - the 3rd out occurred before BR reached first base. (Absurd, right?). No, the out "occurred" back when he swung for the 3rd strike, meaning the third out was actually R2. Run scores.
Same logic in the OP. The out occurs when the 3rd strike occurs, regardless of how long it takes to call it.
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"Many baseball fans look upon an umpire as a sort of necessary evil to the luxury of baseball, like the odor that follows an automobile." - Hall of Fame Pitcher Christy Mathewson
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