That looks to be the wrong wiring diagram, there doesn't seem to be any connections for the lighting on the right hand switch, just stop/run and starter button. Check the wiring coming from the right hand switch, there should be a brown/white, make sure that's connected to the brown/white on the loom as that's the side light wiring.
The wire you think you need looks to be the one way solenoid for the starter circuit so it works when the bikes in gear and the clutch is pulled in.