Ok, the switch has three positions as established in your first post:
Off, self explanatory
Run, also self explanatory, I hope.
Start, this allows the bike to run at idle only. Attempting to rev the engine, this does something to the timing to keep it from reving up, since this is a one-speed "auto" bike, with no neutral starting the bike with the throttle WO could end very badly, for the bike and rider.
This is often a cause of frustration among PW50 owners, since if this connection gets wet, it will short causing the bike to run very poorly. Many people disconnect the lead or a preferable alternative is to insulate it very well so it doesn't get wet.
I would try starting it with ether sprayed in the air box, if you can get it to run this way, you most likely have a carb problem. If you can't get it to at least start on starting fluid, you have an ignition or extreme compression problem.
It's also wise to do a compression test on the bike.
If you can get the bike to start on ether, but won't stay running, have confirmed compression is OK (100+ psi is good, more is better, would probably still run on 50 PSI), have a good spark and have tried the easy stuff like new plug and good carb cleaning, you could have a crank case leak. This is the least likely, most frustrating cause of a no-run condition in a two stroke.
Good luck.