Looks to me like you might well keep that handle where you put it to start it! That diagram on the plate is more or less aimed towards gooks in some other country that don't know their left hand from their right! Have you tried running it (operating it) after it starts, without moving that little lever? Could that little handle be what normally shuts the engine off? It doesn't seem like the engine is choking up, making it stop, because it would tend to stay running, but choking it would get the result you have now!
When you start it, try only moving that lever just a bit, instead of all the way to the right. I think you are doing a normal complete shut down by moving the lever all the way! Moving it just a wee bit might open the choke enough to allow the engine to run.
To check that, remove the air filter down to where you can see the intake side of the carb, and watch how that lever moves things in there. This seems simple enough, but trying to guess what is happening won't get you running and working. Long distance help is good, but in this case-all we can do is make suggestions. That carb/choke/throttle assembly looks strange to me, but try my suggestions. HTH: Rusty J.