The backup light connection is the center one (#7) and the screw lug on the backside is VERY close to the battery charge connection (#4) & the running lights (#3). If the backup lights are on regardless of whether the running lights are on, then the short is to the charge line. I HIGHLY suspect that the short will be found inside the connector where the wiring connects. You can easily check to see if the tow vehicle side is the problem with a
12V continuity light. These can be found at any auto store, or you can use a DC multimeter. The whole idea is that the CENTER contact within the connector on the car should NOT active (ie
12V present) except when the tow vehicle is in reverse. If the center contact seems to be working properly when the trailer is disconnected (ie no power to center contact except when vehicle is is reverse...don't check it with the engine running...safety factor) then the short is most likely on the trailer side, but it still could be a loose contact on the vehicle side that shorts only when you plug the trailer into the tow vehicle. Either way, a good visual checkout of the connections on the backside of the connector should be done before looking elsewhere. Hope this helps