Nord is the name of the race of men that is natural from Skyrim. Forsworn are depicted as the naturals from a country within Skyrim, that the Nords occupied. However, they should belong to the same race of men (Nords). Dialogs in Markarth always talk about the conflict between the Nords and the Forsworn.

My interpretation is that they're Nords too, the problem is that there is no term to distinguish the Nords that inhabit the cities of Skyrim, from the ones that want to rebel. If they're not Nords though, what are they? A whole new race?

Nop, they are Bretons, or a scission of them. It appears in their lore.

I thought the Orcs were the natives of Skyrim, but the Nords pushed them out.

Im pretty sure the forsworn are Breton, apparently they are leftovers from the conflict of bretons and nords.

The Forsworn are Reachmen, which are basically a subset of Bretons. The Nords took over the Reach late in the 2nd era.