What is Time-Varying Voltage?

In circuits, we have no problem saying that the voltage difference between two point is [itex]\cos(\omega t)[/itex], but what does that actually mean? What is "voltage" in this case? I ask because my understanding is that in time-varying electromagnetics, the electric field is no longer a conservative field which can be written as the gradient of a scalar. We have to write [tex]\vec{E} = -\vec{\nabla} V - \frac{\partial \vec{A}}{\partial t}[/tex] i.e. (if my understanding is correct) the notion that [itex]V[/itex] is some sort of potential energy (per unit charge) is no longer valid because the electric field is non-conservative and so it has no associated potential energy.

So I have two questions:

1) When we talk about "voltage" in time-varying circuits, do we mean the scalar field [itex]V[/itex] as it appears in the Lorentz gauge (or maybe Coulomb gauge), or do we mean some other quantity?
2) What is the physical interpretation of "voltage difference" in the time varying case (if there is one)? I'm assuming you can't just say "It's the potential difference per unit charge" or something similar because we're not working with conservative fields anymore.

Additional note: I know that in a lot of practical cases (e.g. 60 Hz household wiring) you can just invoke the quasi-static approximation and continue to interpret "voltage" as it's defined in electrostatics, but what I'm interested in is the regime where the quasi-static approximation is no longer valid (e.g. RF/Microwave).