The regulator: it sounds like it is OK. The Black wire on it goes to the coil, and to the upper relay contact. The Green wire is the ground. The White wire is the current to the alternator's field coil. The Black will always pass power to the Green, that's the normal operation. There is a resistor mounted to the back of the relay: that is the "mid-charge" current limit to the field coil. This is not a "relay" in the normal sense, as the coil pulls the upper contact downward in proportion to the voltage on the Black wire.
The way it works (for your future reference) is:
-When the voltage falls below (about) 12.4 volts, the coil does not pull hard enough on the moving arm. This lets the upper contact on that arm meet the +12 volt (Black wire) terminal. The moving contact, which is connected to the White wire, then receives a full +12 volts to send to the field coil and puts the alternator in full-charge mode.
-When the voltage on the Black wire rises to about 12.8 volts, the coil pulls more strongly on the moving contact and disconnects it from the upper (Black) wire's contact. Now, the current on the moving contact, which is connected to the wirewound resistor on the back of the regulator, is connected to +12 volts only through that resistor. This lowers the current to the field coil, reducing the alternator's output to about 60% of full power.
-When the voltage on the Black wire rises above 13.2 volts (K0/K1/K2 bikes) or 13.6 volts (all later SOHC4 750s) it pulls harder on the armature, which moves further down and connects the White wire to the lower contact, which is the Green wire, or Ground. This stops all current to the field coil and the alternator output drops to about 1/4 amp.
So, it's not really a 'relay', although it looks like one. And there is always a connection (the coil) between the Black and Green wires, which in your situation is enough of a load to light the little 3W bulb (or an LED type bulb tester).
There is often some confusion about this regulator, mostly because the earliest sandcast 750s used one that acted like a relay as it did not have the 3-position arrangement - only FULL charge, and LOW charge. In those, when the voltage exceeded 12.8 volts it pulled the moving contact open, which dropped the field current to 0 (these did not have the big resistor on the bottom). It did not ground the contact, it just left it open so the alternator dropped to 1/4 amp output. These were all recalled early on (in 1969) for new regulators because they made the headlight flicker dim-bright-dim as it cycled, quickly (some called it "strobing" when observed on the hiway at night) and some State Police entities thought it "looked too much like the flashing lights on our Police motorcycles". Touchy bastards, back then....
So, we ended up with a unique, one-of-a-kind, "linear relay" voltage regulator in the CB750, which was never seen before, nor since!