I've always referred to designing how things function as "user experience design." I agree that the terms aren't perfect. The skinning or rendering stage as you call it is still very important. Most of what we point to as good design (on the surface) is because it is well skinned.
I've written two books (http://nathanbarry.com/webapps and http://nathanbarry.com/app-design-handbook) on end to end process, so now I'm working on teaching the practical skills to create the end result all new designers are looking for.
I think we agree about the process though. I always start on paper before moving to Photoshop and code.