I think you are almost there.
The black-red should be the thinnest of the wires connecting to the starter. It goes to the coil and it seems to take 12V during start to the coil, bypassing the coil resistor. (If you short this wire to the red wire then the lights come on as they should because you are energizing them through the resistor as if you had the switch in the ON position).
There is a black wire that feeds the starter relay and should be twice as thick as the black-red but much thinner than the red wire from the battery.
Short story, the black-red wire has no effect on the starter engaging. Leave it hanging for now. You have only two possibilities for connecting the black wire. If you hear a click when you turn the key, then you have it right. And finally if you hear a click but the starter does not move, either the starter relay is bad, or like others said there is too much resistance on the black wire or ground paths. These conditions are trivial to test, try that and call me in the morning...
