Let's move forward, and not look back.
I think that I'm starting to get it.
Play A: A1 ends his dribble, intentionally throws the ball in the air, runs several feet, and catches the ball that hasn't touched the floor.
4-15: Dribble: ART. 3 The dribble may be started by pushing, throwing or batting the ball to the floor before the pivot foot is lifted.
After the player ends his dribble, he throws it into the air. We all know that a player, all by himself, no defense nearby, may occasionally start his dribble by throwing it into the air (the rule says thrown to the floor, but gravity will end doing that eventually). So that's the start of a possible second dribble, and his subsequent catch of ball seals the deal and makes it a dribble for sure, more so, an illegal (double) dribble.
Play B: A2 ends his dribble, intentionally throws the ball off his backboard, runs several feet, catches it, and then dunks.
Now add in the throw subsequently bouncing off the backboard. Fundamental 19 tells us that
a ball which touches the front face or edges of the backboard is treated the same as touching the floor inbounds, except that when the ball touches the thrower’s
backboard, it does not constitute a part of a dribble.
So with no backboard involved in the play, it's an illegal (double) dribble.
But with the backboard in the play, the portion of the play where the thrown ball bounces off the backboard back into the hands of the player is not considered to be a part of a dribble (Fundamental 19) so it's not an illegal (double) dribble, so it's legal.
https://youtu.be/uAskXXKV2GU
This apparently legal NFHS "LeBron James play" has nothing to do with traveling, and/or foot movement, it's all about legal dribbling, and/or illegal dribbling.
The possibility of traveling (running to catch) was the shiny object that distracted me. With one rare exception, one must be holding the ball to travel.
Can anyone please confirm, or deny, the reasoning and logic behind this interpretation?