I think it does depend on the horse to an extent, but I find it very odd that they put a 120lb weight limit out there in general, you'd have to have a very tiny horse or a very sickly horse (and I'd then question if it should be ridden at all by a student at that point; because even a hundred pounds on the back of a horse with spinal issues isn't a good thing.).
When I was constantly riding my POA/Arab crosses who are just over pony size by inches I weighed 170 at my lightest and I am 5'2" . Currently I am 220 and volunteer at a rescue that uses their rehabilitated horses for lessons and they have no issue with me riding and training.
I would look extremely sickly at 120, in fact my goal is to get back down to 170 and that is it. I didn't read through all the posts, but I read through a few and it looks like you found a place to ride, I'm happy to hear that.