This is such a grey area with no specific guardrails.
Your daughter is green. After coming off twice in one lesson, I think a better instructor would have pointed her at something she could successfully do with that same pony, and directed their efforts at being successful for the rest of that lesson and been done with it. A pony or horse that tries to dump a green kid in a lesson is not in the same situation as that same pony that a daredevil kid decides to tackle on their own. Why? Because the daredevil kid decides whether to keep climbing back on or not and where to point him. The lesson kid gets told to get back on and gets told where to point him. That’s just not the same scenario at all.
I am sorry about her wrist. Falls DO happen, and yes, I grew up in the ‘you all right? What’d you do wrong?Get back up there’ camp and yes, for a MORE seasoned kid, there’s a time and place for that. Getting dumped in front of a cross rail twice in a lesson can take a lot of stuffing out of a kid, and maybe she was tired and losing focus and balance and confidence, so the last fall is icing on the cake. Maybe the pony is a punk, who knows, as it can indicate the horse/pony is tired, sore, or what have you.
She doesn’t have to get back on THAT one, that’s fine. By the way, you’re not a helicopter, you just care about your kid.