Hi Steve, it's been awhile!

Sure sounds like the rear window defog switch is running the engine! After activated it stays on that long! (:) just kidding!

You have some good ideas floating out there from other posters but I keep trying to think what would time out like that and ended up with that above.

But seriously I like a loose PIN connection on the AMM, especially, if one gone pushed back up inside the connector.
I too had your thought of a bad relay and again it would have to be an overheating coil that when its gets hot enough to drop out the magnetism. That might explain the cooling off period and so forth.
I'll say it would be a rare duck of a relay if it was but we know about those solder joints.
When I was 16 I had a "generator," on an Italian made Harley-Davidson motorcycle, that had a field winding do that!
It drove me crazy having the idiot light coming on so regularly.
It turn out to be bad wire connection joint, that sprung apart, when the engine got hot and it shutdown the power output. I Resoldered it and all was good from then on!

I was going to ask you to feel the ignition coil for excessive heat, but I have no idea what's too hot?
They do run warm and with engine heat I have no idea past that. Again it might be really rare!
Try putting on some test lights on the coil, like Art showed recently in a post, it might help.

Other than that, I'm stumped too!

Phil