I really think that it is a combination of old and new terminology. Back in the day, when MFWD/FWA/etc. small tractors were rare, the only four wheel drive tractors around were the large articulated tractors, with front and rear tires of equal size. These were, and still are, the four wheel drive tractors.
Since we have entered an new era in tractoring, and the advantages of added front wheel assist to small tractors far outweighed the cost and have become popular, the term gets a little fuzzy. I believe there was a time when John Deere, and maybe others, added hydraulic powered front wheels, for a true front wheel assist, but it was never intended to be used as such for an ongoing basis (like use it only to get through a sticky spot).
After that, the MFWD (mechanical front wheel drive) term came about to distinguish these tractors that had an actual driveshaft going to the front axle as opposed to those that used hydraulic drive.
Since then, I think that these terms have confused the casual CUT buyer, but everybody understands what a 4WD truck is. So in sales literature and seller to buyer conversation, even now the little tractors that have mechanical front wheel drive are called four wheel drive (I think mine even has stickers on it saying 4WD).
Now, are our little tractors with mechanical front wheel drive true 4WD - well, sorta. They have the capability to transmit power to all four wheels, but may not all the time (or may not part of the time due to lack of a differential lock in the front and/or rear). In the grand scheme of tractors past and tractors future, the little tractors with MFWD will probably continue to be called 4wd just so folks don't confuse them with 2wd (I guess). Of course, I had a Belarus 420 that was 4wd of some sort - but it had a viscous coupling and it decided when to power the front wheels. You could not shift it in and out of 4wd. If the back wheels overspeed the front by too much, it was engage the front (and it worked well climbing hills and using the loader). So I don't really know what version of MFWD/4wd that was.
As far as I am aware, a Quadratrac is either a type of transfer case used in 4WD trucks/SUVs/Jeeps, or the Case-IH designation for the type of tractor or tracked tractor with equal front and rear tire sizes. For a long time around here, whenever anyone talked about that type of tractor, it was a Steiger, no matter if it was a Case, John Deere, New Holland, whatever. I would guess it is a regional dialect type thing.
In conclusion, if your tractor can engage power to the front axle, and you want to call it four wheel drive, I don't think your wrong.