Personally, I think the category of medium armor is kind of meaningless. I think there should be heavy armors that are a little lighter and less protective than other heavy armors and light armors that are heavier and tougher than other light armors, as well as armors that are both relatively light and strong for each category. There should be more variation within the armor classes instead of just a straight progression from light and weak to heavy and strong like Oblivion.
It does make sense though to have light and heavy armor governed by different skills, as the use of either one would require a different skill set. With light armor, you retain a good bit of your mobility, so you would focus on trying to angle your body a certain way so that the relatively weaker armor can deflect the blow rather than just absorbing. On the other hand, heavy armor is strong enough that deflection isn't as vital, but it would also make moving much more difficult, so you would have to learn to move within the limits imposed by the armor.
Personally, what I think would make the most sense is to have two skills, one called heavy armor, and the other called dodge or something similar. Heavy armor should have certain penalties in attack speed, mobility etc. which the heavy armor skill would reduce as it increases, as well as somewhat increasing your damage resistance, since being more used to moving in the armor would allow you to absorb blows more effectively. Dodge should more drastically reduce the damage you take both while wearing light armor and while unarmored, since light armor doesn't impair your agility that much, so the practice you get while wearing armor should also help you dodge blows while unarmored and vice versa.