I know what the OP asked. In the literal sense it means A1 lifted his pivot and returned it to the floor before passing or shooting the ball, which is exactly what 4-44-3a says. 4-44-3b is what addresses a player who jumps. Everyone is telling Wade it's an illegal dribble but no one has posted a rule showing it to be an illegal dribble. 4-44-1 most definitely doesn't tell anybody it's an illegal dribble nor does any part of 9-5.

Struckoff (sp??) may be an idiot but complaining about her doesn't help as far as citing a current rule or case play for this scenario.

Yes, it's an illegal play. But the rules book isn't well written. The case book may ruled it as traveling, but the traveling rule is too vague to support the ruling. And some of these folks say that the correct ruling is a double dribble and that the ruling was incorrectly changed by a dumb bimbo.

Dribble
Art. 3. During a dribble, the ball may be batted into the air, provided that it
is permitted to strike the playing court one or more times before the ball is
touched again with either hand.

According to this rule, I pretty sure that's an illegal dribble violation in both situation.

#1, Who said Struckoff was an idiot?

#2, as for the rules, 4-15-1 & 2 are the pertinent rules as it describes the action. Since the release of the ball is not a pass, shot or fumble, that leaves a dribble. When the player releases it and touches it again before it touches the floor, it's an illegal dribble.

#3, finally, the last rule to apply is 4-44.
Traveling (running with the ball) is moving a foot or feet in any direction in excess of prescribed limits while HOLDING the ball.

You cannot travel without HOLDING the ball except as in 4.44.5B which is an identified circumvent of the rules.

In 4-44-3a, the player lifts AND returns the pivot to the floor while still HOLDING the ball.

In 4-44-3b, the player jumps AND returns to the floor while still HOLDING the ball.

You cannot travel without HOLDING the ball.

Just call it, and let the signal be secondary to the call on this issue.

It that were strictly true, the case of a player tossing the ball in the air while not moving the feet would also be an illegal dribble. However, that play only becomes a violation of the player moves their feet between throwing and catching the ball.

I think this case really falls between the two rules (travel and illegal dribble) and is judged to be an unintended action and is arbitrarily lumped with one of the two rules by declaration since neither really cover it exactly.

It's in between the two
 

I tend to agree with Camron, that the situation is in between the two rules -- traveling and double dribble. 4-15-1 defines a dribble as "... bats (batting) or pushes (pushing) the ball "to the floor" and in the case of a player catching the ball before it hits the floor, the basic definition of a dribble has not been met.
Also, 4-15-3 defines the manner in which a dribble is started, and uses the same qualifier "to the floor", which indicates that if the ball does not strike the floor, a dribble has not been started. Thus, to toss the ball, and catch it before it hits the floor, cannot be a dribble, nor could it be a double dribble, to do so, after the player had already ended a legal dribble.
Then, we have the definition of traveling with an integral part of the travleing definition refering to a player "holding the ball." So, a player tossing the ball into the air, after having established a pivot foot, and then moving that pivot foot before catching the ball, must not be a travleing violation, by the strict reading of 4-44.
We know that we would call the violation. But whether we clasify it as an illegal dribble, or a travel becomes a case of semantics.
(Isn't it ineresting how off-season discussions lend themselves to such thoughts and discussions?)

At the heart of this is the history of the "air dribble." If one reads the NFHS handbook, he will learn that for a time period this action was allowed. The player could progress down the court while tapping the ball into the air repeatedly without allowing it to strike the floor.
This action was changed to illegal and "to the floor" is now the important part of the definition of a dribble.

As the game is now played, in order for the players to move the ball from one end of the court to the other, they must either pass or dribble the ball. Why? So that the opponents have an opportunity to obtain it. This is not american football in which players hold the ball and run with it to advance it on the field of play. In basketball, any advancement must be accompanied by the release of the ball permitting some chance that the opponent can get it. Contrast that with when a player remains in one place with the ball. Since he is not permitted to run with the ball, he is governed by the pivot foot restrictions of the traveling rule. If he wishes to move from that location while keeping the ball himself, he must dribble. If he wishes to advance it with the help of his teammates, then he must pass. There is no other method by which a team can advance the ball. It's that simple. A team can always elect to try for goal from where they currently are, but that is relinquishing possession, not advancing the ball with possession.

Therefore, if the player attempts to move the ball around on the court from one location to another, we need simply ask by what method he was attempting to do so.

1. Is he holding the ball? If yes, then he can't advance it further than permitted under the pivot foot restrictions of the traveling rule. Once those are breached, the player has traveled.

2. Has he released the ball? If he tosses it to his teammates, then he is passing. If he releases it to himself, then he must be dribbling, and that dribble must meet the definition or it is an illegal dribble.

Tossing the ball into the air and catching it is only legal per the casebook ruling if the player does not move his pivot foot, and this is precisely because the player is not attempting to advance the ball down the court.

After understanding that background, we can determine that a player tossing the ball into the air, moving to a new location based upon his pivot foot, and catching the ball without allowing it to strike the floor must be an illegal dribble because it is not a pass (definition is "to another player") and the player was not holding the ball and moving with it. Traveling prohibits "running with the ball" as in american football.

Ice Road Basketball ...
 

Good explanation. Didn't I see this on the History Channel?