[Rob1968]

It seems we're addressing several different situations in this thread. When A1 throws the ball and moves his pivot foot to catch the ball before it hits the floor, it is considered to be traveling, 4.44.3 D (b), because the ball did not touch the floor, therefore the toss of the ball doesn't meet the definition of the start of a dribble, 4-15-3. Yes, it is a vague, (actually not mentioned) part of the travel rule, but understood and accepted universally, that even though A1 is not holding the ball, during the movement of the pivot foot, it is traveling, to throw the ball and then catch it after moving the pivot foot, and before the ball touches the floor, or another player, or his own backbourd. Hence, the description of "passing the ball to oneself", is illegal. (Of course, we get into the facet of whether the toss of the ball was judged by the covering official to be a valid shot attempt, but that is yet another adjunct to the discussion.)
When the ball is again controlled by A1, after the ball has hit the floor, the element of whether A1 had used his dribble, before the toss, comes into consideration, in judging whether the actions constitute a double dribble. 7.1.1 considers that the ball hits the floor, inbounds, before A1 again controls it.
Although the following scenario is highly unlikely, it may illustrate the differences noted above: Supposing that A1 leaps in the air, catches the ball that was headed out-of-bounds, tosses it high enough so that after A1 lands out-of-bounds, he recovers and returns inbounds and then catches the ball, before it hits the floor. Is there a violation? In my thoughts, because A1 was airborne when he caught the ball, and threw it back over the playing surface, there would be no violation. However, if A1 had a foot or feet touching the playing surface and tossed the ball high enough so that after his momentum carried him out-of-bounds, he recovered, re-established himself inbounds, and then caught the ball before it hit the playing surface, we would consider his actions to be traveling, as expressed above.
Ah, the minutia of it all . . .