It might not be necessary to “have the other installed”, just to say. The Blizzard launcher will run the WOW Client, but just as an account with BfA and an account without BfA have subtle differences, WOW: Classic likely will. (When we played it as part of the BlizzCon virtual ticket, the splash screen was Classic, the version number was 1.13 instead of the modern client / WOW version. It’s not just log in and have it phase to a Classic world.)
NOTE: I don’t see that anyone linked to Ythisen’s actual response - so here: