What is a dribble

[MiamiWadeCounty]

Some people say that a dribble is merely the act of pushing, batting or tapping the ball to the floor. While others say that it is not a dribble until the ball returns from bouncing off the floor back to the player's hand. Here is the scenario:

A1 has ended his dribble. A1 pushes the ball directly down to the floor. Ball bounces off the floor, but A1 does not touches the ball. What's the call?

Off of my interpretation of the rules, that's a double dribble. But I can also look at this scenario as being a bounce pass or a fumble that cannot be recovered legally (if there's such a thing as a fumble that cannot be recovered).