From my point of view (as owner/operator of an acoustic guitar), the presence of a microphone is completely irrelevant.

The sound of an acoustic guitar, even when played through a mic, is completely different from the sound of electric guitar.

In fact, even the presence of a decent pickup in/on an acoustic guitar does not necessarily make the music less "acoustic." The exception would be a case where the pickup is either sonically very crude, or deliberately designed to distort the instrument's natural/acoustic tone in an effort to masquerade as a Stratocaster, or whatever.

And yeah, yeah, I know that there's no microphone or pickup in the world that perfectly reproduces the pure sound of an instrument. So what? When it's necessary to make oneself heard in a large area and/or noisy environment ~ or simply to record ~ you do the best you can, which is usually passably OK. The sound you could produce when all alone in a sonically perfect room would be better, of course, but that level of perfection is neither necessary nor is it a realistic expectation in the typical performance venue.

I also don't think there's any significance to whether or not the song selection includes exclusively traditional material. The low-key sound and the personal vocal approach of a solo performer (or a small group) playing non-electronic instruments is essentially the same whether the song being sung is new or old.

I hold to this opinion as an aspirant to the traditional role of "songster." If Mississippi John Hurt could adapt popular songs of his time (like "Creole Bell" or "Funky Butt") to his solo-guitar picking style and his customary vocal style, I insist upon my right to do the same kind of thing with, say, "You Don't Know Me," "These Days," and "Do You Know What It Means (To Miss New Orleans)." Not to mention 20th-century songs a bit more likely to be considered by some as "folk," like the works of Bob Dylan, Tom Paxton, Woody Guthrie, etc.

What I do may or may not be what you'd call "folk music," but it is definitely "acoustic music," whether I'm playing quietly at home or through a mic and PA out in the world somewhere.