No.
MLB rule 2.00 defines 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. "
So the answer to your question is no since a foul tip that is caught is only considered a strike.
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The foul tip is called out if caught because it could be in bounds or out of bounds. The umps don't know where it is until it touches the ground, so if it didn't touch the ground, how could there be a fair ruling on if it was out or in. SO if the ball is hit and caught without touching the ground, it is an out, if it looks foul or fair.
According to MLB Rules, Section 2, a foul tip is defined as:
"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."
Only if it is on the third strike would the batter be out if he swung through the ball. Otherwise it would just be a strike. If it was a true bunt attempt and the ball did not go directly into the catchers mitt but was deflected up a bit, the batter would be out on any strike.
Yes. Many people learned that the ball must go over the batter's head, or at least six feet high, in order for it to be considered an out when caught. However Baseball rules only make an exception for a foul tip, which is defined 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."
A ball that traveled in an arc before being caught would be considered a foul ball, and catching it would retire the batter, regardless of whether or not it went over the batter's head. As a practical matter, the case described in the question is very unlikely, so the practical difference between the actual rule of foul tips (sharp and direct) and what many believe the rule to be (below the batter's head or under six feet) is probably irrelevant.
Only on a third strike. If it is tipped and caught at any other time it is simply a strike. Keep in mind that this is only for a foul tip if the ball significantly changes direction in an upward direction and is coming back down then it would considered a pop foul and would be an out if caught.