Does the "Display Composer for Classical Works" setting display the composer's name only if the genre is exactly "classical" or will it also display the composer's name for genres such as "classical (piano trio)", "classical (symphony)" etc.? I'd prefer the latter, but the limited experiments I've done so far don't tell me exactly what's going on behind the scenes.

Yes, it's very simplistic. It only works if Genre is set to Classical.