Jack,
If I understand correctly, your last post explains everything. If you really have a one wire alternator, there is nowhere to connect an ignition warning light. I would expect it to be disconnected and off all the time, but maybe the person who wired in the new alternator connected it somewhere that causes it to be illuminated. Given that it comes on with the headlights, maybe the person who installed the alternator thought they were being helpful by wiring the ignition warning light as a "headlight warning light". With a one wire alternator, your alternator relay is also unnecessary, so I'd expect it to be disconnected. As mentioned previously, you really need to trace where the wire to that ignition warning light actually goes...
-David