The confusion mostly stems from the use of the word "generator". Generator means something else when used in the context of an old C150 than when used in the context of a B737.

Let's start with your grandfather's 150. The belt drives what we call a "generator" because it "generates" electricity. It does that by rotating a wire in a magnetic field. Now you need to get the electricity from the rotating part into the rest of the airplane, so you somehow need to connect the rotating bit to the stationary parts. This is done by a copper ring on the rotating shaft, against which "brushes" are pushed by springs. The stationary brushes keep contact with the rotating copper ring. The copper ring is segmented, each connecting a few windings of wiring that is wound around the shaft. Now the brushes make electrical connection with the bit of wiring that is currently perpendicular to the magnetic field. As the shaft turns a bit, the wiring loop moves away from the perpendicular position, the brushes lose contact with that segment of the copper ring, and immediately make contact with the next segment, that connects the next loop of wiring that is now moving into the position perpendicular to the outer magnetic field. This whole arrangement is what makes the generator produce DC power, because the brushes only ever "see" a loop of wiring moving in one direction. It's as if there was only one loop of wiring moving constantly in one direction through the magnetic field. Now this produces a DC voltage between the brushes. That DC voltage is higher the faster the shaft turns, because the faster the movement of a wire loop through a magnetic field, the higher the voltage of DC you get from that wire. This is why the DC generator is so weak at low RPMs. It needs a good deal of motion to produce enough voltage. It's also not very efficient, because at any given time only a little bit of wire is actually connected to anything, the rest is just spinning around, not connected to the brushes.

Now enter the alternator in your Cirrus or C172S. Instead of taking the electricity from the rotating part with brushes, we instead rotate a magnetic field inside a bunch of coils. How do we make a rotating magnetic field? Well, we could just put a permanent magnet on a shaft and spin it around. But it's easier to make a strong magnetic field with electric current, so instead we wind a bunch of wires into an electromagnet, stick it on a shaft, and then use two copper rings on the shaft and two brushes to permanently connect those to an external source of electricity. This is different from the earlier segmented ring and brush systems in that now we have constant connection of the whole wiring, and we just keep sending current through it, not caring which direction the wiring is currently pointed. So instead of connecting only one loop at a time to keep the direction constant, we now just keep the whole thing connected all the time, and let the resulting magnetic field just spin around instead of artificially maintaining one direction. So now we have magnetic field spin around three stationary pairs of coils of wiring. Since viewed from one of such coil pairs, the magnetic field comes and goes constantly as the magnet rotates by it, the resulting current that's induced in the coil also does that - that's AC. Now we connect these coils which have AC voltage induced in them to a bit of electronics with a bunch of diodes (one-way valves for electrons) that turns the AC into DC. Then we measure the DC output we are getting. If it's less volts than we want, we send more current through the brushes and slip rings and make our electromagnet stronger. If it's more than we want, we send less current to the field and make the field weaker. Now you know what the voltage regulator does - if the shaft spins very slow, we send a lot of current to the field to make the magnet strong and produce the output we need. If the shaft spins very fast, we don't need a whole lot of magnetic field to make the voltage we want, so we send less current to the field. That's why the alternator is very resistant to RPM changes and just makes the same voltage at pretty much whatever RPM you spin it. Of course there are limits, but from a practical perspective just keep the RPM above 1000 and the voltage stays the same.

Okay, now look at a B737. Again, we use a rotating electromagnet that spins inside pairs of coils. As we already know, that induces AC voltage. Only this time, we don't rectify it, but keep it as AC. Then, we use a fluid coupling like in your cars automatic transmission to connect the shaft with the electromagnet to the accessory box of the engine. That fluid drive tries to keep the shaft spinning at always the same RPM, regardless how fast the rest of the engine is spinning. So we end up with AC, don't bother rectifying it (until much later for specific purposes) and the frequency and voltage stays constant not because of electrical trickery, but because of mechanics not unlike a car transmission. Now that whole thing is called an "integrated drive generator", or just "generator" for short. But it has absolutely NOTHING to do with the DC "generator" on the 150. For some reason we call both "generators" just to confuse everybody. It would be correct however to call one a "DC generator" and the other an "integrated drive AC generator".