Find someone w/ a voltmeter, and test the 2 stator wires that's shown, with the yellow meter. Meter on 2000 ohms, the 2 wires should read 450 to 550 ohms. If it is any where near 420 to 530, I would get a new one $25 or so.
Them wires connect to your stator coils, it's a generator, it powers your coil and charges your battery. It's also AC current so, a hot wire goes thru the voltage regulator with makes it DC current to run your lights, and charge your battery. Look at this diagram, it's german, but it gives you the wire colors and were they go in simple form. It shows the eng and the wires coming out and where they go.
That wire from the starter goes to one side of the starter solenoid, the other side goes to the battery I believe. The small wires off of it, think one maybe black goes to the starter button which gets its power from the keyswitch which gets its power from the voltage regulator. There is a red power wire too, right, that might hook to the safety switches that probably ground the starter till you pull the brake lever etc. When you press the starter button it connects the two big wires and the starter turns. The rust is on the outside, just clean it, and the inside surface good.
Have to have one small wire off it to the start button. Put 12v to the yellow w/ red stripe and see if the solenoid clicks. Also with power to the above wire, check to see if there is voltage in the other wire. If it don't work ground the yellow w/ green and see if it works.
Maybe to get power for something else, without going thru the factory wiring. You need the Rectifier to turn AC from the stator coils to DC the charge the battery. The rect. is usually diodes, the electricity goes thru but can't go back like AC current.
My X18 battery has been dead more that 2 years, I just kick start it. My 125cc ATV you have to have a battery. Might want to try simple wiring, tncscooters has all the white plastic plugs cheap, that and alittle wire, you can make you own basic wiring. You can run your lights off the battery, or make a power wire off the Volt. Reg. rectifier to power your other stuff, run it thru a 10 amp 12v. fuse.