The answer to this question becomes complicated because the common use of the term “foul tip” is much broader than the strict rule book definition
By definition,a foul tip is a batted ball, which goes directly and speedily from the bat to the catcher’s mitt or hand and is legally caught by the catcher.
If it is indeed a rule book foul tip, it is a strike on the batter, the ball remains alive. By Rule 8.18.j, it is not necessary for the runner to retouch the base after a foul tip. The runner has legally stolen the base.
If the batted ball travels directly from the bat to any part of the catcher’s body or equipment other than the hand(s) or glove/mitt, it is a foul ball not a foul tip (even if caught). In this case or if the ball is not caught, it is a dead ball, foul ball. A strike on the batter. The runner must return to the base.
If the ball does not go directly to hand or glove and the catcher catches the ball by moving to first touch the ball with her hand or glove, it is a caught foul ball. The batter is out; the ball is live and the runner is subject to be put out on appeal for leaving the base early.
Maryland State Umpire in Chief