"Classic" is different than "Classical". Classical music is a genre and is timeless, just as "Big Band" is a genre and represents, nearly totally, music of a couple of mid-century decades.The music/radio/marketing industries must then operate differently than virtually all others. The use of the descriptor "classic" does indeed refer specifically to the music.
"Classic" in radio can refer to hip hop, country, pop, rock and other music genres. Classic hip hop is can be 90's and 00's, while classic rock can be 70's, 80's and later. Classic Country is often 90's. Classic AC in Spanish is mostly 90's and 00's and early 10's.
Classic all depends on the age of the target listener and the evolution of each music genre it represents.
And in advertising and marketing, Boomers are (except for medications and the like) not a target at all, while those in the younger groups are the "new" targets.
Stations self-define with standardized format names so that time buyers will know what they are. Buyers do not get subtle differentiations as Nielsen has a list of "standard" formats and each station picks the closest one to what they do. If a "new" format comes on the scene, Nielsen will add it to the list if a significant number of rated stations reaching "a lot of people" request such a name be added. I got that done once for "Spanish Adult Hits" and it took about 30 stations owned by at least three groups to get that term standardized.
Remember, many of the format names are never used on the air. They are intended to identify stations within the radio and ad business.