If you attempt to dominate your horse, they see you in a different role than that of the leader. They may comply with your demands but it will be out of fear. Fear of getting hit, fear of the whip hitting them in the face, fear of getting kicked. You are not after dominance of your horse. Your horse does know the difference between dominance and respect.
Stallions fight for dominance in the herd in order to reproduce.
Respect and dominance are concepts that the horse knows far too well. The horse associates dominance with fear. The golden rule really applies to horses. The horse has a good sense of right and wrong, they don't keep score but they do do build long term relationships where respect and trust are the foundations. I want my horses to look at me as if I were someone they could trust with their life.