Basic question about conduction current
Yes, current only moves if there is a change in voltage and flows from a higher voltage to a lower one. There is only one current in this case. 

In the case of a capacitor, current flows in and out of the device only when there is a change in voltage on the terminals, otherwise the current through the device is zero.

In most cases there is a voltage source which creates a voltage difference and the current flows back to the source.

With static voltages (sometimes physical processes), you can get current without a 'circuit' and return flow back to the source.