Personally, I have a slightly different take on this. I don't see myself as playing different roles in different contexts, rather expressing different aspects of myself.
The presentation example is a good one. I'm an introvert, but for more than a decade was a qualitative researcher. So a significant chunk of my job was presentations (pitches, and presentations of findings), and an even bigger chunk was moderating focus groups of 8-10 people with, typically, at least half as many people viewing from behind the one-way mirror. I also frequently spoke at conferences to, typically, audiences of a few hundred.
So that was a job in which I had to be 'on' or 'up' for significant chunks of time. The rest of the work, however, was as introverted as you can get: sitting in a room by myself analysing all the things we learned, and putting those presentations together.
Initially, I thought of the 'up' time as playing a role, and the 'down' time as being myself. But as time went on, I realised that both were me – they were just different aspects of me.