There should be 3 kinds of horses.
a transport horse , normal speed, no armour, no bump capabilty. For getting from A to B faster than on foot scouts, engineers. The romans had mounted archers, but they used the horse only for transport, getting off the horse to engage.
Maybe 2-3 extra slots for carrying extra gear(ladders,siegeshields,contruction sites).
fast mobile light armored horse used mainly for lancing ,HA and such. Bumping with them should do low or no damage, and damage the horse equally.
And armored warhorses , slower ,high armour beasts trained to trample footsoldiers severly with armor soaking up the damage to the horse.
Price according to balance.