A non warhorse would likely do OK in a fight, but if it's struck in combat or something like that, be more likely to decide to balk or run. In game terms, I'd say that a warhorse, on average, should have higher resistance. . .i.e. Will and Stamina, than a riding horse. (Likely higher hits, and some decent attacks of it's own also).
So if the horse needs to make a morale check due to a blast of fire say, the warhorse is more likely to make the Will check with a higher bonus and not bolt than the riding horse. The rider would only need to make a control check if the horse tries to run, if it makes the will check, they can keep fighting without having to pause to control their mount.
Beyond that, for any type of horse, I'd treat training level as a riding bonus/malus, almost like an item. i.e.:
the player gets a just broken to the saddle wild horse, it's -25 to all riding checks, as it's not really tame, hates being ridden, and has urges to run away.
vs
a superbly trained riding horse with a very steady and calm personality is generally obedient and hard to spook, so offers a +25 to riding checks.
Needless to say, the latter horse would cost a lot more. . .and a well trained, "bonus" warhorse with the higher resistance levels would be VERY expensive.