It's 1am, and I'm about to hit the sack, but here's my recollection.

The indicator in question looks like a circle with a filament inside it, and maybe a base. It comes on in both the parking light and headlight position of the switch, which I believe you are pointing out may indicate is NOT the headlights. So if there is some resistance in a contact, it may cause enough voltage drop to throw things off. Makes sense.

The fog light has its own indicator, which is working when that switch is on. It has those three slashes pointed down.

I guess I can try deoxiting the contacts of the bulbs in the back, which is easy. Maybe reseating the connectors at the brain of this - as I recall it is located somewhere in the trunk - maybe near the spare tire well?