AR = Earth, Civilian Research
Details, details... I have to disagree with this, because NAR also decorated the shuttle that took Kirk's party to Spacedock in ST6. No research going on there.
Perhaps "Auxiliary Reserve" or something like that? The Vico
might have been on lease to a civilian outfit because Starfleet didn't acutely need her. The Raven
clearly was on a scientific errand in the opinion of her owners, but Starfleet might have disagreed.
Or if not "Reserve", to imply nonactive/surplus status, then at least the "Auxiliary" part, and definitely not the "Research" part. Perhaps Auxiliary vessels come in a number of categories, sorted out by who's doing the auxiliary work. The USN has the USNS for most of its auxiliary work, but Starfleet might have more players there; "R" might be the code for a specific organization, or then for a region of registration, in this case quite possibly Earth.
Whether the NSP-registered transport T'Pau
had anything to do with science is debatable, too. The Okudagram specified "Vulcan National Merchant Fleet".
Well, with the Relativity's registry prefix of NCV, it's a safe bet that the first C stands for Starfleet. Maybe the A is for Earth, non-Starfleet registry, S for Vulcan, non-Starfleet, etc., with the third letter indicating something about the ship itself.
Like I said, this'll all be worked out somehow into something resembling logic, or at least consistent with what was shown on screen.