PTO typically referred to a shaft for power was taken from a machine or tractor to manually Drive another item or an attachment. A lawn mower cuts grass so the deck isn't actually an attachment it's part of the mower. Once again just a given.
It's also funny how the term pto is used much more for a switch when you have an electric clutch than it is for a lever to engage the blades if you're arguing it is a PTO taking power from the engine to power the deck wouldn't they both be the same and both a pto?
Well, at the risk of going somewhere I don't care to go...
A Power Take Off - PTO - is a term for any of many methods of getting auxiliary power off a machine besides the wheel drive system. Some are shafts that can have a driveshaft attached. Some are belt drives. Some are friction drives - like a flywheel. Some are hydraulic. Many large tractors had belt-drive PTO's, especially older ones before the 3-point hitch. Any PTO has a method of engaging and disengaging, that can include electric clutches, mechanical clutches, gearshift driven from the transmission, hydraulic. My ZTT42 Cub has two pulleys stacked on the engine driveshaft, one for the trannys and one for the deck. The deck one has an electric clutch, making it the PTO. The type of PTO's used would be determined by the intended use and engineering needs for the implements available. A PTO is what manufacturers call all these things.
A lawn tractor/riding lawn mower/garden tractor (take your pick of terms) will have a PTO of some kind to drive the mower deck. Even the machines used only for mowing have a PTO that engages, disengages and drives the deck, exclusive of the ground-drive system. And most if not all of the "tractor-look" machines have a removable deck, so technically it's an attachment, even if it never gets detached.
How a PTO is used depends on the needs of the operator. If, as you say, 99% of the users just mow grass, then all they need is the mower-deck PTO.
A walk behind mower with blades bolted to the engine driveshaft does not need a PTO. And if you want to discuss them there is a forum on this website for them - it's way at the bottom of the forum menu.