Confusion about one of Maxwells equations

[Ax_xiom]

TL;DR

Why does a rotating wire in a magnetic field only cause a current within that wire

So Faraday's Law of induction states this: $$ \nabla \times E = - \frac {\partial B} {\partial t} $$
Or if we write it in it's integral form: $$ \int E \cdot dl = - \frac {d \Phi_B} {dt} $$ which (to my understanding) means that the magnitude of the EMF around a coil of wire will be equal to the magnitude of the change flux through that wire. I get confused in some cases as when we have a generator like this:

Nowhere in the region between the 2 magnets is the field actually changing, so (with my limited understanding) no electric field should be generated.

If we make the coil stationary and rotate the magnet instead, everywhere in between the magnets is experiencing a change in flux, so everywhere should have an electric field, but one is only generated within the wire

So what is wrong with my understanding?

 

[anuttarasammyak]

TL;DR: Why does a rotating wire in a magnetic field only cause a current within that wire

If we make the coil stationary and rotate the magnet instead, everywhere in between the magnets is experiencing a change in flux, so everywhere should have an electric field

Yes.

but one is only generated within the wire

No, loop integral of electric field is generated with no regard to the wire. When free electrons in the conductor wire is on the loop, force which is charge times electric field is generated.