I'm not an expert but...
Stihl was one of the first. I think they started mass production in the late 20's in Germany. They also designed the modern chain based on the way some type of beetle chews through wood.
I don't think American saws really took off until after the war. Lombard is a name you don't hear much anymore; but they were one of the first American manufacturers, started in '43. The oldest saw I have is a late 40's Lomdard. The rest are early 50's to 1972.
SOME SAFETY ADVICE:
I'm sure others do; but I would NEVER use one of these saws for actual work even in the yard. I can't imagine using one in the woods. I grew up on a 200+ acre tree farm and fell trees for firewood all through high school (I'm 26). I've seen people get badly hurt with modern saws. I've seen anti-kickback mechanisms potentially save lives. Also I never pick up a saw without chaps and ear and eye protection.
A lot of fun things are potentially dangerous; but I see no point in risking life and limb, just to lug around an absurdly heavy, noisy, and inefficient saw.
Edit: By the way, I have a Model A with no seat belts (yet) and mechanical brakes. I'm not a prude; but you need to consider risk vs reward.