I once spent about 1.5 hours trying to understand how that system worked and posted it I think on this forum. I promptly forgot the exact details. Voltage regulators and wiper motors are my weaknesses.
I believe the light will only work if there's a difference in voltage between the alternator output and the ACC/RUN voltage in the underdash area. Now before you think "aha!", the light bulb needs something like 8-9 volts difference across the filament to be visible, so the overcharging condition you mentioned would not be sufficient. The resistor wire is there because if the bulb burns out, the VR still needs a stimulator signal to tell it that regulation is needed.
The overcharging condition is due to a bad voltage regulator. Stop running the car now, and replace it with a good quality VR (motorcraft?). An overcharging battery can explode and do lots of acid damage to sheet metal and other things. Double check your wires and contacts at and around the VR, as that may contribute to the VR not working right. Many of those pins are corroded and I often have to replace the pins with new brass ones.