So it is my understanding then, that these currents really do not exist. Do surface currents really exist?
Yes, they exist. Lets look at a DC example:
A different view for an impinging field would be this but the result is the same, because for an impinging field the magnetic field still generates a current and then all the laws that govern current come into effect. It doesn't matter how the current is generated.
Source: Muon Ray
The DC currents taper off to zero at the core because of the internal magnetic field of the material of the conductor. So its a real effect from the interaction of the current and electric field.
As we increase the frequency of the alternating field only the electrons on the surface move. Why? because the core of the wire has a higher impedance. High frequency currents take the path of lowest impedance, which happens to be near the outside of the wire. One thing that would be interesting to try would be to come up with an expression for impedance vs radius.
Another thing to not is the electric and magnetic field outside of the wire, which also carry energy.
If you look below, you can see what portion of the wire is actually carrying current. If you move to high enough frequency, only the outside of the wire carries the current.
Source: Quora