Yeah, I agree with this. I don't see what Other Mom's issue is, unless she overheard something incorrectly, or Stranger Mom reported it incorrectly. All you did was tell someone who had a complaint, to take that complaint to the coach. That seems like a perfectly reasonable way of handling it.
Though now I'm wondering why Stranger Mom even had to point you out--it's not like the coach was hiding his identity and demanded of her, "How did you know I was the coach?! Who told you?!" Why not just say, "Hey, you're the coach for that kid over there, right? Well he was doing XYZ bad thing earlier..."
She probably didn't know who his coach was. The kid wasn't staying in one place, remember?
And the only reason she approached our OP is that the OP linked herself to the boy by admonishing him. The other lady assumed the OP was his mom; she was wrong in that, but she was *not* wrong in thinking that the OP had some sort of link to the boy.
Oh, right, I understand why Stranger Mom approached the OP. But when the OP referred her to the coach, I'm trying to imagine a conversation that really necessitated pointing out the OP. If I had noticed a misbehaving child and was going around trying to find their responsible adult, I would probably say, "Excuse me, are you the coach of that boy over there? Well, he was doing XYZ bad thing. I thought you ought to know."
Well, here's one, assuming the kid had run off by that point. "Excuse me, I noticed a child doing XYZ bad thing. He's run off now, but that person over there [points to OP] indicated he was one of your charges. She can tell you his name." That makes me feel better.
Not trying to be nitpicky, I just wasn't understanding why Stranger Mom felt the need to involve the OP in her conversation with the coach. But it makes sense she might have to if she didn't know the child's name AND he wasn't there to point to any longer.
I was just thinking that maybe Stranger Mom said something that made the OP seem more like a busybody (unfairly), hence Other Mom's otherwise nonsensical comment. I like to try and force these things to make sense, but obviously, some people (like Other Mom) defy sense...