About 1/5th the way down is Article 4 'Vehicle categories', Section 2.a.(ii):
Regulation (EU) No 168/2013 of the European Parliament and of the Council of 15 January 2013 on the approval and market surveillance of two- or three-wheel vehicles and quadricycles (Text with EEA relevance)
www.legislation.gov.uk
> L1e-B vehicle (two-wheel moped);
It really is a moped, with all the laws that come associated with that (inc type approval, reg plate, helmet, brakes on both wheels, etc)
Failure to register one doesn't mean it isn't a moped. For example, if I don't register my motorbike, it's definitely still a motorbike!
I'm now mildly regretting suggesting RichC gets a faster one, but you're forcing me to educate myself, which is definitely appreciated.