The Baseball rules differentiate between a foul ball and a foul tip.
A foul tip caught by the catcher is a strike. If it's the third strike the result is a strikeout, otherwise the at bat continues.
A foul ball caught in the air by any fielder (including the catcher) is a pop out (or fly out, which is scored the same).
Major League Baseball rules define a foul tip as follows:
A FOUL TIP is a batted ball that goes sharp and direct from the bat to the catcher's hands and is legally caught. It is not a foul tip unless caught and any foul tip that is caught is a strike, and the ball is in play. It is not a catch if it is a rebound, unless the ball has first touched the catcher's glove or hand.
Many people were taught that a foul tip is anything that does not go over the batter's head, or that does not go at least six feet high, but those criteria are not in the rule book.
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If a fielder catches a ball and falls into the stands, it is a dead ball and all runners advance one base. Rule 704(c) The fielder made a legal catch. At that point the batter was out. The fielder then took the ball out of play. At that point the runners were allowed to advance one base.
If it is caught by the player in the stands with no interference (such as a fan tipping the ball up), it is an out.