Correct , the 2 outer pins are in & out for the reverse lights so it doesn't matter which wire goes to which side , the brown is the ground for the relay , there is an outer pin & an inner pin on the switch the outer pin is only depressed in reverse , the inner pin is depressed , only in Park & neutral allowing the relay to trigger the starter . The brown wire in the center of the switch should never have power , it is just to ground the start relay , if the brown wire has power maybe there is a short in the harness somewhere connecting with the power wire for the reverse lights .
I hope this helps