It does not need a battery to run. I don't think any of those wires actually need to be connected to get spark.
Off the top of my head, I believe the plug with two wires went to the neutral safety switch. One goes to the electric start button. You can check with a meter which one connects to ground when button is pressed. That wire connected to the starter solenoid. One of the other wires is power for the lights. Should be able to check for ground when ignition on and light switch on.