Don't forget that we do see what a "Class J" classification looks like. See Mudd's ship cargo ship from "Mudd's Women".
An "old Class J Cadet Vessel Starship" = "small Class J cargo ship".
Now, you are kidding? Throughout TOS it had been established that starships are the most sophisticated vessels and the most rare and wanted command opportunities (apparently, back in the 1960's and in the original context the term "starship" meant something else to audiences before "Star Wars" came along. Retcon Alert!). And the oldest one we became aware of in TOS was the Archon
in "Return of the Archons" (maybe a ship like that was a Class J starship?).
Just because there obviously is a similar alphabetical classification system for cargo ships and shuttlecraft (and planets one should add) doesn't mean, that just because there happens to be a Class J starship
and a Class J cargo ship
they are one and the same.