Contrary to the belief of some, trinary is a nonword invented by uninformed computer scientists to rhyme with binary. Ternary describes "three" as a cardinal number (e.g. number of objects or subdivisions, as well as 3-valued number systems by extension), whereas tertiary is the successor of secondary.
There are thus three terms competing for the positions of numerical base, cardinal adjective, and ordinal adjective, only two of which happen to be valid, according to the OED: ternary, trinary, and tertiary.