When is a dribble a dribble?

[Back In The Saddle]

A1 shoots and the rebound is batted around a bit, B1 bats it toward mid-court where A2 finally secures control by beginning a dribble with the bouncing ball, by which I mean that the ball was still bouncing and rather than pick up, then start a dribble, A2 is able to get the bouncing ball under control by dribbling it. However, his very first push (a dribble is begun by pushing the ball to the floor, right?) dribbles the ball on the center dividing line. I call backcourt, then allow the players, lack of sleep and those darned voices in my head to talk me out of it.

Looking back, my question is this: In a situation where a player controls a loose ball by beginning a dribble, where does the dribble (and thus control) really begin? Is it on the first push to the ground? Or does it require a second push to demonstrate that he's actually got it under control?