I've hesitated joining the fray here for a while, but I think I've finally "percolated" on it enough to be able to explain my thoughts. There are a few issues intertwined here.
One is whether we should try to replicate some supposedly "authentic" version of a song or reinterpret it. Obviously, opinions differ. Some people still play Dixieland jazz more or less as it was played in the 1920s. In the dulcimer world, we are lucky to have people doing their best to preserve traditional styles of play as well. My own opinion is that we should study traditional sources of our music, but if we do nothing but replicate slavishly those traditions, then the tradition would be dead. However, even in trying to copy those traditions, we necessarily re-interpret them, don't we? When I was trying to play blues guitar and copied note-for-note what Robert Johnson played, I knew I did not sound like him and had to make the tunes "my own."
If we accept that copying exactly what some earlier performer has done is worthwhile to try but ultimately impossible to accomplish, then we should be freed up to study those earlier versions but make our own music.
A second issue is how precisely we should play on beat. I would suggest that we should all be able to play on beat, but that once we have that ability we need to free our creativity to pause in between lines, hold some notes or chords longer than others, play some sections louder than others, and so forth. I would only insist that those variations should be done on purpose rather than the result of an inability to stay on beat. A couple of years ago I arranged a version of "Ashokan Farewell" and was kind of proud of myself. I posted it at Stephen Seifert's Dulcimer School, and his feedback was eye opening. He praised some of the chords I chose but also suggested that I vary the tempo and the dynamics to create a more expressive piece. He was basically telling me that I was playing on beat but sounded like a robot. If you're playing a contradance, you obviously have to stay on beat. But if you are playing solo, especially if you're singing, then the dynamics and tempo should follow the emotional expression of the song and vary accordingly. At our family reunions for years we would gather around a campfire in the evening and sing songs. My mother was the main singer, and from the age of 10 or 11 I began to accompany her on the guitar. One year I had been trying to play with some really cool rhythms and I got frustrated that my mom would pause in her delivery, or hold some notes way too long, or whatever. My uncle saw my frustration and whispered in my ear that my job was to accompany the singer and not impose any kind of speed or rhythm on her. That comment changed my perspective about what I was doing and how to add creativity to a rendition of a song.
I, too, hear Barbara Allen as in 4/4 time, even though this discussion led me to seek sheet music and several versions on YouTube in 3/4 time. I don't know if anyone could prove that one is more "authentic" than another. But I would suggest singing it however it feels right to you. A big part of maturing as musicians is learning how to play within ourselves. If a rhythm or chord or note or word does not feel right, don't play it. Find what feels right to you.
Finally, I have to confess that I can't hear Joan Baez's voice without crying. The lyrics don't even matter. There is something in her voice that is too pure for this world.