Here is a twist....what if it is a dropped third strike by any of the definitions but the umpire calls the batter out.
This is exactly the reason why the correct umpire mechanic taught on strike three is to only say or signal, "Strike three" or "Strike". We are taught to
NOT add "Batter out" as part of our normal third strike call.
If you routinely say "Batter out" on EVERY third strike, you are eventually going to say it sometime when the batter is NOT out (like on an uncaught third strike). Now you've just created a huge problem for yourself, you are probably going to have an angry coach to deal with, you are obligated to straighten out this mess and, when you do, there's a good chance that
the other coach is going to be mad at you.
This falls under the jurisdiction of rule 10 which requires the umpire to correct any situation where a delayed or reversed call puts either team in jeopardy. Calling the batter out
might cause the batter to stop running when she's not really out. It
might cause the defense to stop playing, or cause them to make a different play than they would have had the batter not been declared out.
The remedy for correcting this is...whatever the umpire judges would be needed to correct the effects of his reversed call. This could involve having the out stand, or possibly placing the batter on first base. It would depend on how the players reacted, where the ball was and all the other circumstances of the play.
It's bound to get ugly and one coach or the other isn't going to like what you come up with! So much confusion that can all be avoided just by following a basic, simple umpire mechanic: DON'T say "Batter out" as part of your normal third strike call!