Under the Sale of Goods Act a trader is the same as a dealer. You have full protection under consumer law.
To prove he is a trader you need to show that he sells several cars a year and advertises them online or with For Sale boards. So you would need to print off any ads you come across or take pictures of his other cars. If he is telling you he will fill in the yellow trader/dealer section this is a warning that he is most likely a trader.
You can check if the car is currently taxed:
https://www.vehicleenquiry.service.gov.uk/
It is not unlawful for traders or dealers to leave the previous keeper's name and address on the registration document. This was originally designed to prevent the history showing a lot of previous keepers otherwise every time a car was part exchanged, the trader would put his name on the V5c and the history would show twice as many previous owners.
You should be very wary of doorstep traders who try to hide the fact they really are car dealers and not private sales. If there is an issue he will say he sold it privately and you have no consumer protection at all.
I was tricked into the same position as yourself then discovered the car was a write off, bought in Scotland, sold to me and found to be totally unroadworthy when I properly examined it. I only got my money back when the police visited him and threatened to charge him with selling an unroadworthy car. The Dept of Transport carried out an MOT and closed the garage that issued the test certificate the previous week. A lot of hassle you can do without.
My advice is to get the deposit back and go elsewhere. If you have problems with him contact Consumer Protection:
https://www.gov.uk/consumer-protection-rights
If he really becomes a lot of trouble you can report him for selling cars from his home to the planning enforcement section of the council where he lives saying he is carrying on a business without planning permission and paying the additional business rates. If he is a trader the Inland Revenue probably don't know about him either.
There are ways and means.