Good morning, Hendypanoply.
(1) I think that all books would classify "baseball" as a noun
in your sentence. It refers to "game." That is, the game = baseball;
baseball = the game.
(2) But in "The baseball game has started," I believe that most books
call this a noun that is being used as an adjective. That is, it tells us
what kind of game: good game/ bad game/ baseball game.
Have a nice day!