A-1 catches the ball while in the air, lands on one foot, jumps and lands simultaneously on both feet. A-1 then pivots. Travel or not?

Travel

Yes, it is a travel.
Once you execute a jump stop (which is what you describe in the land on one, jump to two) you may not pivot with either foot. You may however, pick up either or both feet and pass or shoot before returning the foot/feet to the floor.
Rule ref: 4-43-2a(3)

NevadaRef is correct. A pivot foot can always be lifted, but not placed back on the floor. An easy way to remember it is that a player after dribbling can always jump into the air (with one or both feet) in order to shoot or pass.

Why is this different than the play discussed in the other thread about A1 "alighting", jump stop, and then shooting. Is is because of the "pivot"?

I'm not sure what's in the other thread, but after a jump stop (as used by officials; not necessarily as used by coaches), a player cannot pivot. It becomes traveling when the pivot foot (either foot) is lifted and returned to the floor, or lifted and a dribble is started. It's not traveling to lift the pivot foot and shoot or pass (which, I'm guessing, is what's in the other thread).

Thanks Bob. The other thread is "Casebook 4.33" and Chuck Elias answered my question over there.