My reference to "inbounds or out of bounds" was based on 7-6-1, which states that the throw-in must touch another player inbounds or OOB before going OOB untouched (similarly in 9-2-2). I had incorrectly assumed that the "or OOB" was also included in the definition of when the throw-in ends. But it's not.
I'm sorry if I've confused people. I'm a little confused myself now. How could the throw-in NOT end if it is caught by a player who has one foot OOB?

It's not a throw-in violation, right? B/c the thrower has not violated 9-2-2. Hmmmmm.
So A1 has the ball OOB for an AP throw-in. A1 throws the ball and A3 catches the ball with one foot OOB. We
have to switch the arrow here, right? But if we say that the throw-in doesn't end when A3 catches it OOB, then we can't switch the arrow. So why isn't that "or OOB" included in the definition of when the throw-in ends?
I must be missing something obvious.