You're quite correct in everything you've said. The thing is though, you're missing only one small piece to the puzzle. The model is, as you seem to know already, the base. It's the "physical" structure around which a skin is mapped. No model can be used without a skin, just as no skin can be used without a model.
Here's where the difference comes in though. You could have any number of skins all for the same model. When you call up a particular NPC, for example, it loads a skin and a model, but that does not mean the same model can't be loaded again a moment later with a different skin. Some in-game examples are the Reborn. Notice how they all look nearly identical, but their faces are different as is their "uniform"? That's because all of them use the same model, with different image files to act as the skin.
Does that answer your question?
Oh, and really there's not much a difference between SP and MP models. In fact, there's no difference; it's all in how those models are used. One of the things I'm working on right now, in fact, is how to import some of the models commonly used in MP into an SP mission.