Throw-in player catching deflected ball OOB
 

I've always been kind of confused as I have had officials tell me different things...

A1 under the basket is throwing in the ball. As they attempt to throw it in, B1 intercepts the pass and deflects the ball right back to A1. There is plenty of time for A1 to get out of the way and let the ball fall to the ground OOB, but they catch it anyways while still standing OOB. Does the following throw in go to Team A or Team B?

Also, what if the deflection is so quick that it is a reaction for A1 to protect themselves and catch the ball so it doesn't hit them in the face? Is that different than the sitch above?

OOB on A. Give the ball to B. It doesn't matter how/why A touched the ball out of bounds.

See 7-2 (Causing the ball to go OOB) --especially 7-2-2

When B1 touched the ball, the throw-in ended. This means that if A1's throw-in was a throw-in due to the arrow, the arrow is reversed. Since A1 is a player, A1 caused the ball to go out of bounds when A1 caught the ball. Therefore, it will be B's throw-in. Had A1 not been a player (eg: B1 deflected the ball to the A bench), and an A bench personnel caught the ball, it would have been out of bounds due to B, and an A throw-in.

Even if an A player had to protect themselves for safety reasons, they still caused the ball to go out of bounds. It is not different than the first situation in your post.