This is a very good question. Do I have an answer? Not even remotely! I own a Ford 1510, which, in spite of the turf tires and small size, is defined by tractordata.com as an farm tractor. It has a heavy cast iron frame, diesel engine, gear drive, PTO, 3-pt hitch, and I can see the argument for it as it can do a large variety of farm jobs. Swap rims and it can be fitted with R1 tires and plow, disk or row cultivate a garden.
My John Deere L111, on the other hand is classified as a lawn tractor. The frame is made out of formed sheet metal and light weight rails, it has a Briggs and Stratton engine, hydrostatic transmission and a belly mower and does one thing well: mow the grass. It can also tow a small garden cart, but that's about it. Once again, I can see the argument for it.
In between these two are a vast myriad of machines made by many companies that really blur the lines and definitions. They are smaller and lighter than my Ford, but have the 3-pt hitch and PTO. Many of them come equipped with belly mowers and their smaller, lighter frames make for very decent mowing machines. The hydrostatic transmissions coupled with power steering make for very smooth operation around the tight confines of the yard, yet they can use just about every type of implement known to man, so they are also at home in the field. What are they called?
Tractors, I guess.
Joe