Suppose you have a conductor like a wire with current flowing through it, does it make sense to define a notion of surface charge density for it? So, for example can free charge exist on the surface of a wire with current passing through it be AC or DC.
It's certainly possible for there to be both a surface charge present, and a current flowing through a conductor.
The surface charge might or might not be the charge that is moving to form the current, though.
For example, any time current flows in a wire that also has a DC voltage relative to another nearby conductor, this will be the case.