ok the black wire is definitely the ground or kill wire for the motor.
The Red wire from the engine (the 4 prong plug, which only has 3 prongs in the plug) is the charging side of the alternator for the battery. Now the red wire produces about i think 12-14 volts DC depending on rpm and the batterys charge state.
The orange wire, from the same location as the red wire(from the alternator) is another voltage supplying wire, but it produces 12-14 volts AC which is used for running the lights on your mower.
So here is what i would do.
Run the red wire to a wire on the ignition that is hot( or on) when the ignition is keyed on(or switched on)
so basically the red wire goes directly to the battery but you would have a switch to turn the power on and off.
The black wire should go with the black wire on the ignition if you have the 5 prong ignition which kills the motor.
The orange wire i would run to a separate swith other than the ignition(if you already have on) to have the lights on their own circuit, or just completely forget about the ac wire and run the lights off the battery(better if your at night and the motor dies.
here is some pic's i took
here is the 4 prong plug
and here is the two wires (red and orange) running underneath the flywheel where the alternator is.
and here is the black wire that i believe runs to the kill prong on the magneto
ok so those are the pics of my engine, the wires should be exactly the same as yours on the engine. but quick recap
Red=12-14 Volts DC (charging)
Orange=12-14 Volts AC (lights)
Black=0 volts ground for the magneto to kill the engine
also the thicker wire you see in the pic with the wires running into the engine has a diode which allows the dc side of the alternator to charge, if your missing that diode at all it won't do a thing, but you should be able to see it, it has a shrink rap around it and its usually right next to the plug, but if you don't see it at all(which probably won't happen) but if it does seem to be missing or can't see just get a volt meter and see if its producing and dc voltage, if it was missing it you would get funky readings from the voltmeter or nothing. but you shouldn't have this problem.