To me, Anne is generic. Although it is not widely used as a first name nowadays, as you pointed out, because it was so widely used, very extensively used, for centuries and centuries, it can't escape a generic image for me. I feel the same way about Mary and Jane, but Anne to me is also plain and boring-sounding, and they're not.
I prefer Anne to Ann and very much dislike Annie. I prefer Anna to Anne, because it's not quite so plain and generic.