Sorry fella, caught me on a bad day
In officialdom "Scooter" is a "body type".
This is an accepted classification of visual appearance, It has no legislative, taxation or licencing significance or entitlement.
If your V5 is more than a few months old it will be shown in the body type box or body type/vehicle description. It is being phased out so if your vehcle is on the new P categories it will be motorcycle on your V5.
Body types aren't random words, they are very specific.
There are numerous international agreements and documents about this, they aren't universal and they have changed over time, ISO 3833 is probably the best known international vehicle classification standard. If you ever registered or imported a vehicle in the UK you might have the old version V355/1-5 (instructions for completing V55) lying around which has the body types on it and points you at the available definition. It won't show up on google now because the new P codes have come in. The DVLA staff (or DOT etc depending on how far we go back) and manufacturers also needed to know when to use "scooter" so documents were issued to them at various times defining body types, shouldn't be hard to find one.
Then there are alpha numeric codes for the body type used by various allied countries and manufacturers, designating "scooter" as a body type and accepted wording for homologation agreements etc.
All these things vary in detail but the one constant is that the scooter body type does not have a direct structural link between the seat base and headstock ie it has a "step-through", even if you can't actually use it to step through (underbone isn't recognised as a bodytype). Some mention engine and transmission or foot supports, some don't but they all mention the frame layout.
What about the exceptions?
Not my job to worry about them any more so I don't care - I said there was an official definition, I didn't say it was right or that I agreed with it......... However:
The word "headstock" isn't being used in the generic way that google throws up first, I'm pretty sure Irev knows that already and was just being playful.
Body type is a visual category so how direct is direct? What about non-standard body forms?
What about them - it isn't black and white and somewhere one type has to take over from another and it isn't going to be clear cut every time.
Its a bit like saying somebody is a man or a woman, some people aren't either, or they are both. It works as a general rule but not every single time.
"Scooter" was used as a body type for historic reasons and it is sometimes anachronistic, it is also subject to change or abandonment, just as the "jeep" body style got dropped when copyright issues popped up.