What causes the "Generator" light to come on?

[67ChevyRedneck]

I'm starting to get a little peeved at my 67. It's been having some serious electrical problems. For a couple weeks it was randomly overcharging/undercharging. But it wasn't often. It got bad this week, so I changed the voltage regulator last night. Now it's undercharging between 10 and 11 volts and the "generator" light has finally come on (and is staying on.) I took the truck to advance to have the alternator tested. Everything checked out fine This is driving me nuts!

I've already checked and cleaned all of the grounds, the voltage regulator seems to be working properly, since the old one was allowing under and overcharging???

What makes the "generator" light come one? It never came on all the other times the truck was over and under charging. The first time it came on was right after I swapped the voltage regulator.

Is it just a generic light that reads a low battery or is it directly linked to the alternator to let you know that it's bad?

I'm really banging my head on the wall on this one fellas