Throw-in player catching deflected ball OOB

[jeschmit]

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?