I do not know a whole lot about the 044 440 models, were never that popular in my neck of the woods.
However I do know the part number you list for the flywheel is correct for perhaps the earlier period of the series run (at some time they changed to 1214 standard and 1215 Arctic), there was a technical bulletin to support all that- but I am not about to go through boxes of paperwork to find it!
Pretty sure there was a 1200, 1200A and a 1200B but I cannot off the top of my head tell you what if any differences there were between the three.
Looking at your photographs though- that cylinder looks WAY too clean in relation to the flywheel and surrounding areas of the other photos and maybe this saw has been made out of a parts bin. Could be a mixture of old 044, newer 044 and even some 440 for good measure (tank assembly and flippy cap fuel- screw in oil cap).
That is an elastostart handle- the busted one, never seen one do that without breaking the rope first....... if it is a Stihl factory one and not a knockoff copy of a Stihl one. But you want to replace that with an OEM and ensure the recoil is clean, lubed and in good working condition before you try firing it up again.
I know when my 460 bites back, the 70cc range of Stihls can certainly make you take notice! Pulling that broken stub and having it bite back- just looks painful!