BTW, the factory schematic indicates that this wire is brown-yellow. There are also 2 black wires from the NSS but those are for the back-up lights.
If you have an ohmeter, you can check the NSS for proper operation. Check resistance with one ohmeter lead to the brown-yellow wire at the upper end (in the engine compartment) and the other to chassis ground/battery -. It should be near to 0 ohms with the trannie in park or neutral, and very, very high resistance (essentially infinite, or an open circuit) when in any gear.
It sounds like the G terminal indicated on the starter relay has been grounded; otherwise, it would never start. You need to remove any such ground when you hook up the brown-yellow wire from the NSS.