Would you differentiate, when recounting the night to friends, if you drank some bourbon + some rye on a night out? IDK how things are done, in America.
Over here, in England, most people I know wouldn't, because they're all just US whiskey.
We would, however, differentiate drinking Scotch whisky, from bourbon (even while called whiskey).
All I know, is that Rye is a place close to McLean's hometown & The Levee was a bar in his hometown.
So it makes sense, to me, for him to be singing about a time when the bar was dry, so they drove to a neighbouring area where they had booze.
Edit: Just read that, much like rye's 51%, bourbon has the same req for 51% corn mash & it has to be aged in new charred oak barrels.
Both seem to have specific things that must be met, in order to earn the name.