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Player control is established by holding the ball or commencing a dribble. An "air dribble" consists of holding the ball, tossing the ball without it striking the floor, and once again holding the ball.
Holding is not necessarily two handed - Dr. J used to always pick up the ball with one big hand and swoop in for the finger roll or the dunk. So one handed holds can occur, also with a hand directly under the ball carrying it. If you believe that the tap is really a one handed hold followed by a toss, you have an air dribble. But if the tap never starts with a clear one handed hold, it is simply a tap and by rule, you have no player control. No player control, no travel.
Slider, if your extreme scenario were to happen, the hand would be under the ball in such a way that you could call it a one handed hold, and then you have a travel in my book. But if it is tapped above the head, as in a rebound, you would be hard pressed in almost all normal situations to say that the tap was actually a hold and throw. If the ball was to visibly pause in the hand showing clear control, and then the player throws it, that is different than a tap and rarely occurs.
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