Back to the original question, the ceiling of the 737-200 can be either FL350 or FL370, depending upon the type of cabin pressure controller installed, along with all the supporting paperwork from Boeing, etc. Two of the airlines I've worked for had -200s, and we could go to FL370 with one, but the other was only good up to FL350.
As far as aircraft types go, all the positiong data comes from FAA's aircraft situation display (ASD) which uses (in its "raw" form) FAA radar data, including the data tags with FAA contractions as far as aircraft types. The treip.com uses the FAA versions, i.e. B733, B734, etc. for respective -300 and -400 variants, and my guess is that some other tracking sites "dumb down" this aspect to a simplistic "B737" for their customers in order not to confuse them.
ALL views, opinions expressed are mine ONLY and are NOT representative of those shared by Southwest Airlines Co.