The British firmly believe RdR and KoP are the same, see the National Fruit Collection listing. I usually treat them as synonyms for that reason, until there is firm information to the contrary emerging.
Re: the genetic info above, I don’t at all understand what the “x 1700” vs “x 82” means now, but the 1700 one appears to be the real thing based on the genetic analysis. The fact that germplasm from France has a 100% match with the KoP sourced from the UK is another sign the two names are synonyms.
My RdR came as KoP from Southmeadow 20 or so years ago, it is probably identical to the KoP (and the x 1700) at Geneva.