Starter turning on while engine is already running

The only way this would happen is if there is a short to power on the starter control relay connected to the switch for the wiper. So we need to test for that. They way to do that is test for resistance or continuity between the white/pink wire at the starter and each of the wires on the multifunction switch.

You do this by putting your meter on ohms and disconnect each of these connectors and touch one lead to the terminal of at the starter, and the other to each of the wires on the connector at the switch. You should have no resistance reading (open circuit which is technically infinite resistance) or no continuity.

https://www.2carpros.com/articles/how-to-use-a-voltmeter

If you do then those two wires are shorted together somewhere. At which point I would just try and run new wires rather then trying to find the spot and repairing it.

Let me know what questions you have. Thanks