Here is a dissertation I wrote for the Lancia board on alternator issues. It is a thought experiment from reading circuits and playing a bit with a bad Lancia alternator which I replaced with SFDon's help. Now you can tackle your red light on your own and ignore the shop...
In general the alternator rotor is excited by the current going from the battery through the charge bulb in the dash just to get some magnetic field before the alternator is producing any. The rotor is also excited by the output of the alternator itself so once it starts it self-sustains without pushing too much current through the bulb.
When the alternator voltage is above the battery voltage both sides of the bulb are at equal voltage because all involved diodes are conducting. The light should be OFF. If the light is OFF all the time it may be that the bulb is burned or that the rotor coil is open. The bulb wire by convention should be the only BLUE wire in the engine bay, so one can disconnect the blue wire (on the side that is not the alternator) and connect it to the battery. The alternator rotor is now excited even with a burned bulb and one can drive the car. If the rotor coil is open this will not help. An open rotor can be diagnosed by grounding the alternator side of the blue wire by observing the bulb does come on.
The voltage regulator is always a suspect when things do not work right, that is because it can interrupt the rotor current on its own, that is how it regulates (by modulating the average rotor current). I think one can briefly bypass the voltage regulator at low revs to diagnose things, namely to confirm a new one is needed.
The battery is used just for starting the engine and from then on it is the alternator providing the current, but maybe there are some subtleties there I miss.
A bulb that stays on solid while the engine is running could be explained if some of the diodes are shot open (as in not conducting). I guess a bad Voltage Regulator could also cause that if it operated erratically (sometimes charging the battery and sometimes regulating well below).