In "Where No Man Has Gone Before," the word "rim" is never used. It's referred to as the "galaxy edge" once, and otherwise they just speak of leaving the galaxy or being out of the galaxy. "By Any Other Name" uses "rim" once (by Rojan, who isn't a native speaker of English) and "edge" once (by Kirk). "Is There In Truth No Beauty?" uses neither word, referring only to the barrier and to being outside the galaxy.
So the location of the barrier is referred to as the edge of the galaxy twice and the rim only once, and more generally is treated simply as the divider between the inside and outside of the galaxy. Not to mention that "rim," like most words, has more than one definition. It can be used to refer to any edge or boundary.
Besides, it's pointless to fixate on vocabulary over practical sense. We know for a fact that the galaxy is 3-dimensional, that the nearest face of the disk is dozens of times closer than the nearest "rim." If the Enterprise had been assigned to travel past the edge of the galaxy, why in the hell would they travel 25-30 times farther than they had to? They'd have to be idiots to do it that way.
So the location of the barrier is referred to as the edge of the galaxy twice and the rim only once, and more generally is treated simply as the divider between the inside and outside of the galaxy. Not to mention that "rim," like most words, has more than one definition. It can be used to refer to any edge or boundary.
Besides, it's pointless to fixate on vocabulary over practical sense. We know for a fact that the galaxy is 3-dimensional, that the nearest face of the disk is dozens of times closer than the nearest "rim." If the Enterprise had been assigned to travel past the edge of the galaxy, why in the hell would they travel 25-30 times farther than they had to? They'd have to be idiots to do it that way.