Where did you get this from?

Reference, please. I'm not saying you're wrong, just that I don't recall reading that.

The overthrow went to the OD batter who picked it up and tossed it to F1.

Or do you suppose the catcher was backing up the plate instead of being at it?

J/R, Ch. 11 (eleven), sect. II (two):
"An authorized person is a person who is not team personnel,
but whose presence on LBT (or in the dugout) is authorized by
home team management. Examples of such persons would be
security guards, bat boys/girls, ball boys/girls, mascots,
photographers, etc. [3.15] [NFHS 1-2-3]"

My earlier post misleadingly suggested that authorized person interference can result from unintentional touching ("the ball hits...").

Thanks.

What would JR do if someone (relief pitcher / catcher) in the bull pen accidentally picked up a fair batted ball (thinking it was foul)?

Assuming that they belonged to the offense, that's covered under "Interference by an Offensive Teammate." The infraction occurs when a teammate:
1. "blatantly and avoidably hinders a fielder's try to field a fair or catchable batted ball or thrown ball."
2. "intentionally hinders or impedes a fielder's try to field a fair or catchable batted ball or thrown ball."
3. "physically assists a runner."

The penalty is that the runner being played on or assisted is out.

References are 7.09(e) and 7.11.

It's a very good question how to handle that sitch if the DEFENSIVE bullpen grabs a live ball!

Do you think the catcher would move after a loose ball? How big is the foul territory on this field? How fast was the ball moving after being deflected? You don't know where the catcher was or how far from the ball he was.

By picking up the ball the offense has deprived the defense of an opportunity to make a play. (i.e. hindered or impeded)

Really? He didn't throw anything to anyone. The instant he possessed it, play was dead. Period. Everything that happened after that didn't happen. (If you're going to go with "assistance" - which doesn't exist by the way --- then would you allow pitcher to then throw to a base and retire a runner? No ... of course not. Play was dead. This mysterious "assistance" didn't happen.)

Good luck. May you succeed where other's have failed.