Yes there are; it's not just size, it's wheel format and stability, battery type and safety, among other things. All train operating companies have full information on their websites. If a scooter user wants to take it on the railway, it has to comply with safety regulations and has to be approved for travel in advance. If the user wants to use a laptop or eat a meal in their scooter, it should be one that caters for that; I've seen them in use while travelling so don't tell me they don't exist.
The conditions for the use of that wheelchair space state that the passenger should occupy the scooter. This is to be fair to other passengers on crowded trains, but many times it's overlooked because the train isn't that crowded. This time the choice was between the space being used for prohibited goods carriage - the unoccupied scooter - and a passenger in a pram needing the space. The passenger wins over the goods carriage. Would you have it any other way?
My very extensive experience of the PassengerAssist service is that it's generally provided what I asked for, with some notable exceptions which resulted in Great Western Trains (as was) sending me 80 miles home in a taxi from Reading, no less than three times. They also sent me a gold pass first class travel voucher on each occasion.
In my year and a bit living where I do now, I've found the service from East Midlands Railway people up to scratch and very helpful, especially in turfing people out of my reserved seat. YMMV.
If your service from PassengerAssist isn't up to scratch, tell them so, complain, and complain each time. You should at least get your fare back. ToCs hate paying compensation, nag them until they get the message and you get compensation. Passenger help is a statutory obligation.