First distinction[s] should be:

If the device puts Direct Current [DC] out on its terminals, it is a generator.

If the device puts out Alternating Current [AC] then it is an alternator.

By extension a "brushless" motor like an "ECM" will put out AC.

A "Brushed" motor will put out DC.

Lots of misuse of these terms here and elsewhere. Many folks call the dual rotor alternators as seen built here "wind gennies" or "wind generators" But they are actually alternators

Just a couple points I hope help in clearing it up some.

Good luck figuring it out, it is not that hard but can be confusing due to the common misuse of terms.

Tom

The Truth is the Truth, even if no one believes it; and a lie is a lie even if everyone believes it

From then on I loose the plot completely.

By pm generator I assume you are referring to a machine with permanent magnets on the stator. There is no dc on the rotor. If it is a brush machine then the ac on the rotor is rectified to dc on the brushes by the commutator. The windings are ac just as in the exciter of the brushless machine that you describe at the start.

You can put a permanent magnet on the rotor of a car alternator in place of the wound rotor and it becomes brushless but you loose ability to regulate it.

A pm brushless servo motor is nothing less than a permanent magnet 3 phase alternator.

If you fit a 3 phase rectifier it becomes a dc generator. If you fit an active controlled rectifier that acts as the equivalent of the mechanical commutator then it will run as a motor.  Mosfets or similar commutate the current to the windings in a sequence dictated by hall sensors so the thing basically runs as a synchronous motor whose supply frequency and phase is dictated by the magnetic sensors.

There are several ways of making brushless alternators with controlled dc excitation. Any pm alternator is brushless anyway. The next generation of car alternator will probably be brushless but it comes at considerable increase in complexity. The scheme that you describe for power station alternators is too costly for small vehicle alternators.