I have always played that it takes a combat action of the rider to make a non-warhorse do anything in combat, but if the rider makes a Ride roll then a warhorse uses its own combat actions.
So, a cavalry horse, riding horse or nag acts as part of the rider, using the rider's combat actions. A warhorse, on the other hand, can use its own combat actions as long as it has been directed by the rider (Ride roll).
All in my opinion, of course.