Does voltage require a circuit?
Refer to my previous answer: Is voltage the speed of electrons?

Voltage is best thought of as a field. We're used to thinking of gravitational fields being entirely uniform, but magnetic fields are not. If you attach a piece of ferrous metal to a pole of a magnet, it extends the field into it. Similarly the electric field between the two poles of a voltage source can be extended with electrical conductors.

This extension of the field extends all the way to the field inside the field-effect transistor.

(Knowing about fields clears up a lot of misconceptions brought on by thinking about electrons. Ignore the electrons.)

Edit: so the explanation is almost exactly your (1) with one detail different. Dirac16's hint is important. The gate-to-channel insulation has a capacitance. So the circuit looks like a capacitor. These two circuits are equivalent (ignore the values):







simulate this circuit &ndash; Schematic created using CircuitLab

So there is no steady-state DC current flow, but at the point of connecting up the circuit the capacitor charges, and while it is charging a current flows. This is actually quite important when designing power MOSFET systems: you need to be able to supply adequate current for a very short time.