Single wire reverse light switch for 68 Barracuda

That is the dual function switch for mid 68 and later. It does neutral safety on the center pin and reverse on the outer 2 pins. So now the numbers on your trans are a factor. At the same time you said you had a single wire to the trans. You said the reverse lamp switch inside the cabin did work. You could have a later model trans installed or the later rooster comb for that dual function switch installed in the earlier trans. Sorry transition years and parts changing causes so much confusion. My on history.... Our first 67 had single wire NSS in the trans, reverse lamp switch inside on the column. When i put the later model trans under it, I ran my single brown wire to center pin of dual function switch. Changed nothing about reverse lamp switch or wiring until the little plastic actuator pin sticking out of my steering column broke. At that point, I simply routed the 2 reverse lamp wires out and down to the 2 outer pins on that switch in the trans just like factory did to have reverse lamps again.