I believe your own horse is the one you can bond!
I ride a mare of a client, she rides her 3/4 times a week, I ride her 2/3 times a week! The mare knows who is her owner, my client bonded with her in a way that I haven't (it's not my own horse so I don't pay as much attention as to my own), that's what happens with school horses, they are ridden by a lot of people qho don't pay much attention so they don't bond as easily!
Hope it helps
One’s own horse. Of course you can love a school horse, but so can many other people who ride that horse. It’s a much different feel to have a bond with a school horse rather than one’s own. When you have your own horse, you’re focused on that horse solely, and that horse is solely focused on you. The attention from both sides is complete to the other, not one horse with many riders. It’s a chance to bond with that one animal, whilst that animal can trust you as you’re their only rider and focus as well
Your own horse. Though I will say I'm currently riding my trainers one horse who she will occasionally use for lessons. And we do have a bond. However I am his main rider and he is only ridden by anyone else maybe once every month or too. I also spend time with him outside if lessons and riding. I have the opportunity to do much more with him than an ordinary "lesson horse"
your own horse.
you have the ability to bond with your horse, and get to know your horse. the horse knows you and the horse knows that you own it.
whereas, with a school horse, they get ridden by a mulitude of people, and they have lots of attention by lots of different people. They take affection different with someone’s personal horse
Hope this made sense at all!