That wire, green with white stripe, is a trigger for the Delay Relay, and daisy-chains to the high speed fan relay. If you ground that wire, the fan will go into high speed, and the delay relay will trigger the timeout (10 minutes or so) before the fan shuts off.
That's in a perfect world.
Since the high speed fan turns on the instant you start the car, the delay relay is defective. Nothing new there.
And, if you're seeing voltage at that pin, this confirms the delay relay being defective. As I've said forever, remove that relay and your fan trigger problem will cease to exist.