Handling wrong info
 

R2 on second, one out.

B1 hits a routine fly to F7. R2 takes off immediately after the hit, and after F7's catch, is easily out on the appeal throw to F4. Of course, R2 thought there were 2 outs.

The BU owned up by informing everyone he told R2 there were two outs. As the PU, I determined R2 would not have run with the correct knowledge, and put him back on second with 2 outs.

Is there an ASA rule that backs up my decision?

Possibly 10.3.C if you consider this an umpire call.

It's really not a "decision" or "call" that's covered by 10-3-C, in my humble opinion. This is something that the runner should have checked on with her base coach.

Suppose a batter asks the PU, "What's the count?" and the PU responds with, "2 and 1," but it's really 1-2. Then the batter swings and misses at the next pitch in the dirt that gets by the catcher, and she doesn't run to first. The catcher runs back, retrieves the ball, and throws to first to retire the batter. Are you going to let her stay up to the plate and disregard the third strike?

I dunno; maybe I'm wrong, and missed counts and missed number of outs are covered by 10-3-C. But in this play, the third base coach should have been the one informing R1, "There's only one out! Tag up on a fly ball, etc. etc."

No. The official ASA position is that the teams (with coaches and scorekeepers available to assist) are responsible to know the game situation, including outs and count. If they have no book, that is their choice, but not our concern.

The umpires' responses are as a courtesy, and do not serve to change the actual situation, nor to protect either team if inaccurate.