starting a dribble

[rsl]

A1 passes the ball to A2, and A2 knocks the ball to the floor and then catches the ball with both hands. Can A2 legally start a dribble?

Variants:
(1) A2 controls the ball with one hand and pushes it to the floor, then catches it with both hands.

(2) the pass is high, so A2 jumps and knocks the ball down and then recovers it with both hands.

This is just a silly question from a pickup game, but in a real game I would want to see real control while controlling the pass to call it a dribble. But, in the pickup game it looked more like (1) and A2 got an easy layup because his defender didn't think he could still dribble. This is probably HTBT, but what do you guys think?

4-15-1 A dribble is ball movement caused by a player in control who bats
(intentionally strikes the ball with the hand(s)) or pushes the ball to the floor once or several times.

4-15-3 The dribble may be started by pushing, throwing or batting the ball
to the floor before the pivot foot is lifted.