Wires for reverse light

My diagram shows similar wire colors but they are not exact to what you listed. Do you have a wagon or a sedan?

The yellow / gray wire, pin 2, is for the left back-up lamp. The ground wire for both sides is brown, pin 5.

Be aware that this circuit has been needlessly seriously over-complicated by adding multiple unnecessary computers to do things they were never needed for before. Their driver circuits are usually not able to handle the additional load of added light bulbs, as in trailer wiring or other add-ons. I suspect your back-up camera simply needs a voltage signal to tell it when the car is in reverse. If that is the case, there should not be sufficient extra current to bother the computer module. Be careful to not ground any wires as that will do damage to the module. Make your connections while the ignition switch is off. Since the module is still running in case the brake lights are needed, I used to recommend disconnecting the battery negative cable, but that is risky now too. Doing that, especially on European imports, can cause multiple computers to lock up resulting in a very expensive tow to the dealership. This is fairly common just from people replacing the battery. If you don't know for sure what will happen, don't risk disconnecting the battery.

All spliced wires should be soldered, then sealed with heat-shrink tubing. Auto parts stores have some with hot-melt glue inside to seal out moisture. Don't use electrical tape because that will unravel into a gooey mess on a hot day and can leave wires exposed. Scotch-Lok quick connectors are another real bad idea because they don't seal out moisture. Those will result in corroded wires and multiple intermittent problems.