Authenticity is something that fascinates me, though I admit what really gets me interested is the concept of authenticity in the arts (especially in music).
When talking about authenticity one can mean almost anything.
If you were brought up by a stay at home mum and a bricklayer father is it "authentic" to your upbringing to choose to work in the building trade and have a traditional family set-up? What if you're really good at school and love philosophy - is it inauthentic to go to Oxford, or is it inauthentic to settle for the building trade like your dad?
It seems to me that truth and honesty are the important things, and "authenticity" is much more about how others perceive you as opposed to anything that can be pinned down.
It seems to me that people who self describe as living authentically tend to be people who claim that their self perception matches their feelings (eg, "I am a man and I feel like a woman - by putting on a dress I am living authentically").
In my view "I am a man and I want to wear a dress" is the attitude of someone who is infinitely closer to meeting the definition of authenticity.