Since current through capacitor (between the plates) is displacement current, what is the current flowing to the leads of a capacitor called?

Since there is no conduction current through the capacitor, what is the current that flows between the battery and the leads of the capacitor while the plates are charging (or discharging) to their respective polarities? Is it still conduction current even though [conduction] current is not completing the circuit "through" the capacitor? Or is it more like a transfer of charge between the battery and the capacitor plates making it more like drift current? I realize that these are not free carriers, but I’m struggling to reconcile the fact that there is no "physical current" through the capacitor (although displacement current current acts as a "real" current inducing a magnetic field), and the fact the I tend to think of conduction current as requiring a closed loop (ie, conduction from battery + to battery – terminals).

Is a closed loop equivalently created (or simulated) as far as the battery is concerned by the collection of electrons from the positive plate on the positive terminal of the battery and the expulsion of electrons from the battery negative terminal to the negative plate of the capacitor making it "look" to the battery as if electrons flowed all around the circuit?

I apologize if the question is not clear. I understand displacement current between the plates of the capacitor (insulator between the plates, so no conduction). I just want to better understand the physics of the current flowing between the battery terminals and the capacitor plates.