I can certainly understand your confusion over this issue. Forty-years ago, the line between a lawn tractor and a garden tractor was much easier to see. Today, that line has been blurred by the douche-bags in the marketing segment of several companies. One of the smudges was the term "Yard Tractor" and to this day, I really have a tough time figuring out what differentiates a Yard Tractor from a Lawn Tractor.
I don't disagree with what you have been told so far is this thread. By and large, Lawn Tractors are designed and built for light duty work such as cutting manicured lawns, blowing snow, ploughing snow with a blade and pulling a cart or lawn roller. Generally speaking, most of the Lawn Tractors weigh in at 400 LBS or less. They use vertical shaft engines that employ a V-belt to send power to the trans-axle, whether that is a gear type, hydrostatic or some sort of mechanical variable speed drive.
Most often the rear wheels are 10" diameter rims or less and the front rims are 6" or less. They have some sort of "tow plate" at the rear that is simply a hole drilled in a horizontal piece of steel the sticks out from the tractor frame.
At the other end of the spectrum is the true Garden Tractor. It normally uses engines with horizontal crankshafts so that both ends of the engine can be used to provide rotating power. Everything about the garden tractor is built much heavier than the lawn tractor because it must be able to withstand the flexing that will take place when it is put to uses that would cause the frame of the lawn tractor to crack and fail.
Garden tractors offer either a sleeve hitch or a CAT 0 three-point hitch so that ground-engaging implements that are configured to either of those standards can be easily attached. These include several different types of earth turning ploughs, disc type harrows, planter/fertilizers, tool bars, adjustable spring-tooth harrows for weeding, rototillers, bush hog style mowers to cut tall field grass and so forth.
I don't know how ANY manufacturer today has the balls to call something a GARDEN TRACTOR if they do not offer a CAT 0 hitch to go with it, either as standard equipment or as an option. Just look at any tractor in the sub-CUT or CUT segment and see if any of those do not have a 3 point hitch as standard.
The glory days of the Garden Tractor are over with. There are very few true GT's on offer today, compared to 30 years ago or more. The price jump between the GT and the sub-CUT is narrow. Back in the 60's and 70's, you either went with a GT or you pretty much had to make the leap to the much larger AG tractor that was often far too large for most properties. The CUT and sub-CUT's have bridged that gap and the big advantages of having even a Limited CAT 1 hitch is very appealing to the savvy buyer.
Don't be fooled by some of the "garden tractor" claims made by certain manufacturers today. If they don't offer a three-point hitch, then they have no right to call their machine a garden tractor. It's just a glorified, over-priced lawn tractor and is usually being sold on the basis of a high HP rating. Look at the list of attachments the manufacturer offers for their tractor.
If there is no 12" single furrow turning plough in the roster, then they don't have a true garden tractor. Look at the operating weight of the tractor with absolutely no attachments on it. If it does not weigh at least 750 LBS, then it's a lightweight wannabee and not a true GT.
A garden tractor needs ground clearance and 12" back tires are the minimum size needed to traverse rough ground along with 8" fronts. AG bar tread tires can be easily found for 12" rims but finding them for smaller rims is a challenge.
Lastly, PRICE POINT is the big red flag. Entry level lawn tractors can be had for under two grand and better quality models will cost you upwards of 5 grand. But a true GT will begin at about the $8000.00 mark before you start adding options and attachments.