1987 Mustang alternator issues

Do you have those two black wires with orange stripes coming out of a rectangular plug on the side of the generator's case, toward the rear?

You didn't mention having the cluster removed, but yes, the warning light passes current to the voltage regulator for its "wake-up" signal. You can't just hook 12 volts to that point because the regulator is going to ground it in an attempt to turn on that bulb. With nothing to limit current flow, the regulator can be damaged. Once the generator is up and running, about 6 volts will appear on the white / black wire. That voltage tells the regulator the system is working, so it applies 12 volts to the light circuit to turn the bulb off. In case the bulb burns out, there is also a resistor in parallel across the bulb socket that will pass enough current to start the regulator. If you're going to jump 12 volts to that wire, do it with a small light bulb, not just a piece of wire. If the wire colors are the same as early '90s models, that's the green / red wire.

What got my attention was the mention of two black wires. If your output wire is bolted to the back of the generator, you have nothing to worry about, but if you have two output wires in the rectangular plug on the side, that plug is supposed to NEVER be removed. Replacement generators come with that plug already installed and you are instructed to cut the wires and splice them to the car's harness. Some of these generators can produce up to 80 amps and it is ridiculous to expect 40 amps to go through each of those connector terminals but that's what the engineers gave us. Unplugging that connector degrades the tension on the terminals and introduces a tiny bit of resistance. Current flowing through any resistance causes heat to build up, and heat promotes further degradation of the terminals and more resistance. That's why it's real common to find these systems with burned up wires and generators. Quite the design, huh?

The two wires are in parallel to share the load. When one terminal develops a little resistance, the other wire has to handle a larger percentage of the current. That also leads to more heat buildup.

Yup. When you turn on the ignition switch, it puts 12 volts on one side of the dash light. The other side of the bulb is grounded by the voltage regulator. That turns the bulb on. If there were no resistance, (bulb), in that circuit, the 12 volts would be grounded directly and either blow a fuse or more likely damage the regulator. You can use a small bulb in that circuit or almost any small resistor, but the only '94 diagram I have is in my head and it's for an Explorer. I can't imagine there would be a difference in the instrument cluster based on engine size, but if that's the case, we'll have to figure out where to connect that wire to get your generator working without discharging the battery when the engine is off.

I have to run out of town for a few hours. When I get back, I'll check for your replies, then I'll hunt for my diagram I used with my students who were diagnosing the Explorer. I have one that explains how the system works and how to troubleshoot it.

See if this makes sense. It explains how the system works.