And my point is that if you start nitpicking a perfectly understandable (if slightly imprecise) and widely-used terminology, you better not open yourself to the self-same criticism lest people would justifiably ignore your opinion.

After all, what C has is "the C library". It's what standard literally calls it (it doesn't even explicitly provide for existence of other kinds of libraries) but everyone calls it "the standard library". Is it correct? Arguably, no.