The simple, straightforward answer to the opening post is that among most musicians, classical, folk, jazz, or what-have-you, if you hear someone refer to a chord by nothing more that it's letter-name ("C"), they're talking about a major chord "C major").

But if the discussion is about a variety of chords, major, minor, augmented, diminished, or generally messed with, being specific about what flavor of chord it is tends to avoid confusion.

In general, "C" assumes "C major."

Don Firth