The armor icons are drawn on top of the horse's HP bar when the horse has more than 10 hearts. This also affects creative mode.

Screenshots

Survival mode

Creative mode

Found a way to move armor up when player is riding an entity using hasVehicle() but still finding a way to detect if it's a horse with more than 20 health

added a check for if player is riding, and if the mounts health is over 20, armor will move up then, in both survival and creative