[greymule]

Pitch goes sharp and direct off the bat to the catcher's mitt and rebounds into the air halfway to the mound. F1 dives to catch the ball but instead knocks it toward F2, who catches it in the air. If the ball never hit the ground, you'd have to rule this a foul tip also. Meets the definition.

No, because a foul tip must be caught by the catcher, by rule.

But F2 did catch it: ". . . F2, who catches it in the air."

And I agree that sharp and direct to the catcher's hand or mitt cannot be a fair ball, even if the actual contact is made over home plate. (Good luck selling that one, anyway.) Naturally, a ball not sharp and direct off the bat, perhaps spinning upward and backward, could be contacted over the plate and thus rendered fair.