So, in the "buyer" name space, it has both of your names? Sorry, I can't tell from the screenshot.
I started looking into this because it's an interesting question. One thing for sure -- the guy who sold the horse to you is totally out of the picture now, and you won't have anything else to do with him.
My best guess, from what I can tell, is that you two co-own the horse, and that won't change when you turn 18.
Do you have a bad relationship with your father, so you're expecting to have to take legal action against him to claim the horse? Assuming you DON'T have a bad relationship, then for clarity when you turn 18 just ask him to write up a document assigning any ownership of the horse over to you.
PS this doesn't constitute legal advice. And things can vary depending on where you live.
"Saddle fit -- it's a no brainer!"" - random person