Hi,
Good lord, this is an interesting one.
I spent some time going through wiring schematics and was only able to find one correlation between the two components. There is one ground that seems like it could be related. I attached three pics below. The first two show grounding locations for the lighting. Pic 3 shows a ground location for the starter relay. (ground 104 lower left of the engine) If you notice, that is the same ground used by the left headlamp seen in pic 1. Do me a favor. Check the condition of that ground. I have a feeling (and this is a theory) that when the light is in, somehow the power is back feeding and causing the quick engagement. That is where I would start checking simply because of the correlation between the two components.
I have to be honest, the grounds on these vehicles are somewhat poorly designed. When checking it, confirm that it is in good condition. Move it around to see if the connector isn't half broken. Also, make sure there are no corrosion issues.
Here is a link you may find helpful:
https://www.2carpros.com/articles/how-to-check-wiring
Let me know what you find or if you have other questions.
Take care and Happy New Year.
Joe
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Thursday, December 31st, 2020 AT 5:10 PM