After receiving a throw-in from A-2, if A-1 fumbles the ball, is it correct for the official to rule this as the start of a dribble?

When a player receives a throw-in and then fumbles the ball, the action of fumbling does not meet the criteria for initiating a dribble. A dribble is defined as a series of controlled bounces of the ball by a player who is trying to advance the ball up the court. Fumbling the ball implies a loss of control, meaning the player is not in possession of the ball in a manner that allows them to dribble.

For a dribble to begin, a player must have control of the ball, which is not the case when the player fumbles it. Thus, the official's ruling that this situation does not constitute the start of a dribble aligns with the rules of basketball regarding ball control and handling.