Boston, I didn't see this thread until you linked it from the other one, but my thoughts anyway.
Looking at what you have I would suggest that it started out life as an S-10 and then was modified with some 08/08S parts.
The top cover is obviously off an 08/08S or a TS 350, you can see that it has originally been used with the choke slider as the paint is worn off around the slotted hole. Are you sure that the wire isn't attached to the carb? It would make sense if this was an ad-hoc choke mechanism that has had to be frigged with the wire out of the hole that is there to access the LA adjustment screw on an 08/08S or TS 350 because the carb is set up for a rear handled machine which looks like the S-10 or would also fit a TS 360, both rear handle machines rather than the top handled 08/08S or TS 350. I can't think that there even is any difference on the choke between the two setups, but may be the guy doing it thought that it had to be pulled from behind. That said I've noticed a metal flap like you describe near the flywheel on my Contra, is it the kill switch? Does the metal flap have a spring attached to it to hold it ON the flywheel?
How do you choke the saw to start it?
I think that to make the top handled throttle work the carb has had to have its throttle lever changed so that it was actuated from above by a solid metal throttle rod, rather that the cable arrangement used on the TS 360, and I suspect the S-10.
The muffler seems to me to be the biggest clue. I can't see how the S-10 muffler would end up on an 08/08S as well as the wrong carb and an S-10 oil tank. It is possible, but I think you have an S-10 repaired with a new, to it, top cover and a modified carb.
Dan