Typically, the term tractor was applied to that equipment that was capable of doing such work. I'm thinking old IH Cub Cadets, JD 200 and 300 series, Wheelhorse, etc. Typically with heavy cast-iron transmissions, frames that were not made of stamped metal, and typically 5-lug rear wheels, though I seem to remember seeing 4- and 6-lug samples at one time or another. Don't overlook PTO's, hydraulic hook-ups, and category 0, 3-point hitches which were optional on some models.
Yes, they had belly-mounted mower decks. Remove the deck, install a set of wheel weights or suitcase weights, and a set of tire chains, or better, lug tires, and you have a small tractor. Capable of plowing snow, pulling various implements, and other types of work. I have even seen loaders on some of these small tractors, though it was rare, and almost never seen on newer models.
Oh, and don't forget -- most of these old garden tractors were equipped with single cylinder engines in the 10 to 18 horsepower range.
Do not overlook the value of these little tractors. I knew of more than one farmer that would permanently remove the deck on one of these machines and use it for the small, mundane tasks around the farm. They were cheaper to operate, maintain, and repair than larger tractors. They also were able to go more places than larger machines due to their compact size.
4-wheelers have pretty much replaced the garden tractor on farms and ranches. There are a few tasks that ATVs cannot perform, as they are not particularly suited for low-speed, high-torque applications over an extended period of time.
I guess this is a good time to point out the difference between lawn tractors and garden tractors. This is where the marketing comes in. The tractors I described above were garden tractors. Lawn tractors are simply known as such not just due to marketing, but also to distinguish the configuration of front-engine mowers from rear-engine mowers. They have stamped frames, smaller tires which are mounted directly to the axle, and generally have lighter, weaker, components.
Today, you will be hard-pressed to find a true garden tractor at places such as Home Depot or Lowes. TSC has a few, and, believe it or not, but Sears has one or two models suitable for this purpose. The father in law just bought a Craftsman GT5000, and I have been pleasantly surprised at how relatively well-made it is, though I still prefer my Cub Cadet. Oh, and to get a good Cub Cadet anymore, I found that you have to go to TSC or an independent dealer. The ones at Home Depot are cheapened versions.
Not everybody needs the bulk and expense of a $10k - $20k compact tractor, when an economical $2k - $4k garden tractor will suit their needs perfectly.