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.
All true but mainly by definition.
I'm not convinced the term PTO even has to mean on a moving tractor or piece of driven equipment. You could have an engine mounted to a table that drives a saw blade through a PTO. That would take out the exclusivity of having or needing a ground drive system to have a 2nd system powered by a PTO.
Of course decks are removable and by definition it is attachment as why I agree all you stated is technically true by definition. Many/most people consider the deck as part or the machine and it's a given.
The transmission is removable too but no one calls it an attachment. They assume and expect it to be there as they do with deck.
Manufacturers use terms very loosely although in PTO they are by definition correct, just not what a lot of people expect or picture when they see the term PTO. Many think more of the shaft or belt drives to items you often hook up and unhook to a machine like you mentioned. Things like pull behind decks, generators, water pumps etc. I don't know many more than that as I am a city boy who uses riding mowers. Country guys use tractors a lot.
Toro, Hondas, JD and others all make walk behind mowers without the blade bolted to the "driveshaft"/ crankshaft. JD and Honda use a clutch and Toro uses a belt and idler.
This by definition is a PTO system but manufacturers rarely if even use PTO to refer to any part on these machines. Odd.
They call a lever or Button switch a PTO often with no mention of PTO switch or engagement lever or switch. They never call the cable that pulls clutch tight or moves idler to tighten belt a PTO cable but riders do. Again odd.
Is not the system of the belt, shaft, idler etc the PTO and not just the thing that engages it??
Not like a manufacturer doing something makes it right or correct but sometimes they do ,maybe by accident, get it technically correct.
Just observations. It is not consistent.
And that walk behind section....It's there but few people end up there posting. I see all new posts and not many pop up.
Today was one about a JS
John Deere with running speed issue. I was happy!
My preferences are as I started out. I don't like the word PTO being used for a deck engagement unless it being driven out the front, rear, or side of a machine I don't like the work tractor use for a residential grass cutting machine that is sold in a department store for under 2500--lots under 1500.
My wife doesn't like the word moist although a great cake is very moist and by definition the word is 100% correct whether she likes it or not.
I will call the next thing in front of her that has high water content "tractor"
"These brownies are really tractor" Then I will get to rant all about our discussion here.
"Pastor Buddy OUT!"