Just thought I'd document some of the bits that had me stumped - perhaps someone else will save some time and frustration through my findings.
The mystery of the wires coming out of the B&S. Here is what I know:
Intek 344cc 11.5HP Model: 21R877 0056 H5 200915ZD
This B&S does not have a built-in rectifier, nor did I need to buy one. Instead it uses a simpler method of a diode on one of the alternator (stator). There are two wires coming out of the alternator. Both are AC current but one of them has a diode, turning the AC into a very simple & lumpy DC current. Use this one to feed power back to the battery! The non-diode AC lead is used to power the headlights (useless for me as I don't have headlights).
Grey - Carbi Solenoid (anti-backfire valve) to switch fuel on/off. Needs 12 V to open solenoid/fuel. Connect to terminal 'L' on switch. Black - Kill Switch wire - connect to ground via switch. Connect to terminal 'M' on switch. Red - DC from alternator/stator. Should have a bulge for the diode. Connect to battery via the terminal 'B' on the switch. White - AC from alternator/stator. This is for lights. I don't need this on my mower.
There is also a thinner wire from the starter relay/solenoid, activating/closing the relay so that battery power goes to the starter motor. Connect this thin wire to terminal 'S' on Switch. Of course the heavy duty wires from the positive terminal of the battery and to the starter motor connect to the heavy duty connection on the relay.
And lastly, the heavy duty negative from the battery needs to either go to one of the mounting bolts on the motor or to the chassis somewhere.