I would attempt to confuse you with ASCII art, but I'm sure someone else will have a nice drawing.

Or the field moves through (across) the wire.

In an axial flux set-up, the electricity is generated when the wire(s) are between 2 magnets, not when they are over magnets. The field is flowing between 2 adjacent magnets, correct?

Correct.  The field is between the magnets on opposite rotors (and through the magnets, through the rotors, and back through another pair of magnets and across another gap.)  This field is dragged through one side of the coil (and the return field through the other) as the rotor turns.

Larger magnet wire means more amps. The reason we don't use larger gauge wire on generators is that enough turns for the correct voltage physically wont fit in the space available, correct?

Ed

Dual rotor - juice flows when a coil leg is directly over a magnet, making it between 2 magnets on opposing rotors. The field flows in a straight line between
the 2 rotors.

Single rotor - the oposite. Electricity flows when the coil leg is centered between
magnets. The field is an arc between 2 adjacent magnets.

From there, Electricity flows through the wire as it passes through a field in the
direction dictated by the right hand rule (been a while since physics 101). If a
section of straight wire passed through a N-S field while a different section passed
through a S-N field the 2 would cancel out. The coil works because the wire turns
180 degrees before encountering the other, opposite field. From the wire's point of
view, both legs of the coil are passing through an identical field. As it moves to
the next set of magnets the 2 fields are again the same, but opposite, thus creating
the AC.