This one is kind of easy. "Kind of" because its long.
You need to have the same horse for most of the main game. If you pick up a random horse, either because the original is stuck in a place or dead, and tie to a hitching post, the count resets.
You can even have your original horse die, but as long as you dont tie another one, you can still get the trophie. But if you get the exclusive low honor horse, the count will reset. Either be a good boy, or be a bad one from the very start and stay like that.