The thing with motorised wheelchairs and mobility scooters on trains(buses for that matter) is that they have a verification scheme. At the end of which you are given a pass for that particular wheelchair/mobility scooter.
They require written documentation from the supplier as to what battery(ies) are on a particular wheelchair/mobility scooter. Change it, your pass is no longer valid and they can refuse to allow it onboard. You also have a brief test to take. I know of no such scheme covering e-bikes, legal or not.
They have not, there are verification schemes but they are for the rider, to make sure that someone without legs isn't suddenly going to walk and therefor claim ''perks'' that person is not entitled too.
Coming from lived experience (my wife needs an wheelchair) she or we have never been asked for said documentation, but even if we did we did acquire it for the airport, then you get an piece of paper that says what batteries it has, since there is no seal on the battery casing it doesn't ay a lot. and there are 301 wheelchairs that look exactly the same that ship directly from China/aliexpress. We now have an much more expensive one and although that brand is much more reputable, the issue still stays the same the battery isn't really sealed so we still don't known if someone replaced the cells and if someone did how capable he was. (alltough i think you have to trick the software too with the expensive one)
But the point is s bit the same as looking for medical issues i always prefer to do that in dutch, not because i don't understand the englsih part but in general dutch websites including those from the government don't tell me just ''you have pain in you left pinkie, call the Ambulance immediately'' but they first give the advice why you need to go to the hospital directly and give explanations about what would happen if you don't go.
I think that's more of the way to go with these things because this ban comes 20 years to late so it's better to setup an system that allows people to gain knowledge, options etc. etc. so that we don't need bans like this anymore because poeple know how to make batteries secure and/or how to check if they are safe.
DO mobility scooters use Lithium batteries??
Yes, not all of them but most of them have lithium batteries. Expensive brands like Pride still offer lithium batteries as an option but that is to increase their profit above from anything else.