To my mind:-
Anything which needs a tool or key to move it is fixed.
Unless it has wheels, over 18 kg is fixed.
As to installation and equipment, I would add the words current-using, so the installation old supplies current to current-using equipment.
In the main, the installation does not need any special skills or permissions to work on it. At least for the testing and inspecting, so anyone who has passed his C&G 2391 should be able to say if safe or potentially dangerous.
But with equipment, we have items even in the home, which needs specialist skills, be it the central heating, a microwave, or a smoke alarm.
So with the inspection and testing of in-service electrical equipment, we have two exams, one for the administration, and one for the actual testing and inspecting, and it is down to the administrator to engage people to inspect and test, so a refrigerator may be on a maintenance contract, and the vending machine for clear reasons, not every electrician has the key. Remember same rules apply if domestic or industrial, and to remove an item from service means it no longer needs testing, but we do need some method to quarantine equipment not in use.
This is why our electrical compound had to be locked, so no unauthorized person could remove it. Clearly cutting off the plug stops it being used, but when the law requires a moulded plug to be fitted by the manufacturer this process must be stopped.
I do not, any longer, use installed radio equipment, however when I did, the room it was kept in had to be locked, to comply with licence conditions.
The same may apply with guns, so we can't take it as read, that an inspector can go where he likes, so there needs to be again a method to combine documents, so all is inspected and tested, and recorded, even if not all done by the same person.
So we have the equipment register, have you ever tried to keep one up to date, I have, near impossible, and there is always something which drops through the cracks, extension leads would normally be grouped in with equipment, but not current using, but also not installed.
Is a whole caravan a bit of equipment, so should it be PAT tested? The batching plant I worked on was portable equipment, it was on wheels, and could be moved place to place, OK needed 22 tractor units to move it, but it was portable, so in the home, the washing machine is portable i.e. on wheels, but the drier is not!
If not looking at the law, it does not matter, all electrical stuff needs inspecting and testing, does not matter if the installation or current using equipment, and in industry the electrical department will decide who does what. And if it is easier to test the hand drier in the loo when doing the EICR then that's how it will be done, even if the records kept with the equipment register.
But domestic not so easy, and also what is done with a service contract? So my freezer with a service contract, is only likely looked at when it breaks down, but the boiler, likely there is a visit every year.
Also, what is tested, the safety officer may test smoke alarms with a smoke generator, but he likely does not test the supply.