In the International Tables for Crystallography A, table 4.3.2.1, the trigonal lattices are identified as having space group IT numbers 143 through 167. However, only space groups 148, 155, 160, 161, 166, and 167 have $R$ in their Hermann-Mauguin symbols, the others have $P$ in their HM symbols. All symbols have $3$ or $\overline{3}$ in their symbols, as is required for trigonal symmetry. Section 4.3.5.1 claims that the current 'P' HM symbols were a replacement for an early symbol 'C'-which indicates a "double rectangular C-centered cell", and that the replacement was done "for reasons of consistency", but this doesn't actually illuminate the situation very much.
If I were to attempt a definition of $R$ in the HM symbol, I'd say all groups with a $3$-fold or $\overline{3}$-fold rotational symmetry about a body diagonal, without a $4$-fold axis about a primitive vector. But this appears to contradict the usage in the International tables.
So what does the $R$ indicate?