Çok güzel anlattınız"binmek" does not mean "to ride". Turkish and English are different regarding this.
"sürmek" means "to ride" "to drive"
"binmek means "to mount" "to get on" "to board"
So you are not necessarily riding the horse or the bike yourself when you mount one.
You can say the following:
At sürüyor: She is riding a horse.
Atı sürüyor: She is riding the horse.
Motora bindim: I have mounted on the bike. (it's not clear whether I am driving it or just a passenger)