Lots of different types of "regulators". For the one in the photo, the relay or column on the left is the cutout. It connects the generator to the battery when the generator voltage reaches a set amount. If the voltage goes below a set amount then the relay opens and disconnects the generator from the battery otherwise current would flow from the battery into the generator and drain the battery and overheat the generator. That relay has a thin wire winding that is the voltage part and closes and opens the relay depending on the generator voltage, not the battery voltage. The generator is disconnected from the battery everytime the engine is shut down because the cutout relay will open. That is why the advice that you have to polarize the generator when the battery is disconnected is nonsense, it is disconnected electrically all the time when the engine is idled or shut off.
The thicker outside winding on the left is a current winding that helps hold the cutout relay tightly closed once the voltage winding first pulls it closed. The cutout relay senses voltage from the generator, not generator speed or battery voltage. Some generators, 3 brush generators, may have only a cutout relay. The current limiting is done with the 3rd brush position.
The relay on the right is a combined voltage regulator and a current limiting relay. The voltage is regulated by a thin winding on the inside of the relay and inserts a relatively high ohm resistor into the field circuit to decrease the voltage or grounds the field to increase the voltage and thus controls the output. It does this by switching the resistor in and out of the circuit hundreds of times per second which is the origin of the term "vibrating regulator" used to describe this type of regulator. It is a mechanical version of a modern pulse width voltage regulator or PWM. It tries to keep the voltage of the generator within a small set range, unrelated to battery voltage, so it is a true regulator.
The other part of the right sided relay is a thicker winding which is a current limiting winding. It will insert that same high resistance into the field circuit if the current output of the generator is above a set amount. All it does is limit the current and does not regulate in any way. The voltage and current sections can be combined into one physical relay as in your picture or maybe two separate relays in others. Combining the two gives less accurate voltage regulation at certain current levels so the combined units typically, but not always, will bypass the current regulator with the "L" terminal. That way the voltage regulation is more accurate. The combined regulators with the "L" terminal are often called tractor regulators. The 3 unit regulators do not have that problem and all current will go through the current limiting relay. The 3 units regulators have 3 separate relays, a cutout relay, a voltage regulating relay, and a current limiting relay.
There are also two unit (Delco) or two charge (Autolite) regulators that function like an automatic version of the charge regulating switch that is often part of the light switch. Instead of the owner regulating the charge by setting the switch position and inserting a resistor into the field circuit, the 2 charge unit inserts a resistor automatically. They will have two relays, a cutout relay, and a voltage relay.
The battery, ignition, and lights receive the output of the generator but they are not needed for the regulating circuit. The cutout and any regulator section sense the generator voltage and output not the battery voltage. Any load will work, does not have to be a battery.
A 3 brush generator depends on a load to limit its output and the magnetic field in the generator will automatically adjust to prevent overloading. They do not need a current lmiting relay. However, the 3 brush generator needs some sort of a load, ignition, battery, lights, whatever but some load at all times. Otherwise, the magnetic field in the generator will shift such that a larger current will flow through the field winding and can burnout in a matter of seconds. The field current can go from a normal 2 amps to over 20 amps in a few seconds with the load disconnected. Not related the battery itself, any load will keep a 3 brush generator under control.
2 brush generators do not have that problem