This may be an extension of my jazz sensibilities, but I tend to favor the musician injecting aspects of his or her own personality into the music, whether its baroque or bebop.
Edgar does this, and given his background, musical tastes, etc, it's easy to understand why he prefers to play the Eccles with little vibrato. (BTW, he plays the first repeat nearly entirely on the D string, I suppose for the darker color).
I've never heard Gary Karr's version, but I would imagine he uses a lot of the gorgeous vibrato and singing tone he's famous for.
Each approach, in my opinion, is valid. A definition of an artist might be someone who takes the information, processes it, and spits it back out with his or her own unique and, hopefully, beautiful stamp on it.