IIRC grated works better because you get more surface area (of potato), and because it's easier to cover the surface of the burn.
Also - when the potato gets warm/hot you'll want to replace it with a new batch.
your good !
What it does is it sucks the trapped heat out from under the skin so replacing it really helps
I remember a kid getting burned from the glass of the fireplace at my mothers house, it worked like a miracle, although the child was still in pain.