It's not ableist to label a phobia a phobia. I have one of sharks and have worked on it with exposure therapy. Sharks can be dangerous to people, but a phobia is a fear that is irrational or excessive. My fear isn't irrational since I do swim in the ocean at times, but it's excessive because the actual danger sharks pose to me is low. OP's fear of horses is a phobia because it's excessive. One bad encounter with a horse doesn't change that.

OP is NTA but if you have kids, you have to teach them about reasonable vs. unreasonable fears despite your own traumas. If the friend hadn't actually put the kid on a horse and had just taken the kid around her horses my judgment would be different. And OP should probably let her kid ride a horse sometime in the future, with proper safety equipment and supervision. OP would not be the right person to supervise that activity either to be honest, because her anxiety would be a problem.