Philip,
You are correct, You can for example push a car downhill with a dead battery on, get in gear and it will start when the engine start spinning a little bit high. But let's take a deeper look.
The generator has magnets inside and it is able to generate electric current fron a zero input, to make it simple. Also note that it does produce straight current. That is, the energy output is adequated to be delivered to the battery and to all the accesories in the vehicle. Note that it is a very good way to produce energy. Even some welding machines uses this devices as current generators to make welding possible.
The alternator is a li'l different. At first it has no magnets, so it must be excited to start generating some current. If they are not excited, You can spin then as fast as You want, they will produce nothing! This needed excitation is put into the rotor, which transfer or induces electricity in the outer windings. The only hassle is that it output is alternate current, just like the one we can find in the plug at the wall. So it needs to be rectified in a diode bridge, an arrangment that converts the alternate current into straight current, just what a car needs.
Nowadays almost all alternators are self regulated, in a way that they work as a stand alone item. Very reliable and very confortable to be installed in an older vehicle. Not to mention that they are smaller, easier to fit and much lighter than a generator.
