A1 is dribbling in the backcourt. B1 "bats" the ball out of bounds still in the backcourt. If I understand the rules correctly the 10 second count starts over for Team A because there is no Team or Player control during a throw-in and the 10 second rule states(not verbatim) that "no team or player shall be in continuous control of the ball in their backcourt for more than 10 seconds".

Please let me know if I am interpretting this correctly or not. THANKS!!

You're correct if you're playing by National Federation or NCAA rules...