When you can't rent a horse, they aren't in the yard, it means that it is Horse Station World Change. Wild Horses spawn is every 3h from ss. Then 0 to 3 monsters spawn each time, this is random.
bestiary gives 10 charm points when you killed 5 horses.
You can tammed it by Sugar Oat or Music Box to get mount and achievement "Lucky Horseshoe".
Btw you should make bestiary for all others horses, it's 5points for each, you must kill 250 but it's easy and don't waste your stamina.