Maggie:
I consider myself to be a bit of both.
In my life, everyone who knows me, also know that I have GID and I am recovering from it through the wonders of medical science and surgery.

Most of everyone who has known me in the past knew me as that other guy. So in that respect, I could not "go stealth". I was not about to sever all connections with my past and start a new life as a deep stealth woman. I am out in my life. I am proud and I am an activist.
I personally feel that the days of people having to "die" in their old life and construct a new life as a woman were done primarily do allow that person to have a productive life. See Lyn Conway's website. She was a very successful engineer for IBM and was summarily terminated when she came out. She spent the better part of a decade rebuilding her life as a deep stealth woman so that she could get a job as an entry level technician and rebuild her life from there.
But because of her efforts and the efforts of transsexuals since then I have been given the ability to transition in my life. No lying, no starting over. Just transition. Starting as the guy and over time becoming the person I have always been, Sandra.
Now to the person on the street, they see a woman. Very few pick up or have a clue that I was not born with a female body. And I prefer to keep it this way. I don't need to wear a sign around my neck that says "TRANSSEXUAL!!!". But if someone does read me I have no fear of admitting that I was born a male bodied female.
Actually, Maggie, you have accomplished much of the same. You have transitioned in your life. You did not have to "die" and start again somewhere else. There was no need. And I think that the acceptance you received from your customers is most touching and most telling of the attitudes that exist today. Not the fundy right wing religious zealots who think that all trans women are men in dresses and pedophiles wanting to invade the ladies rooms and attack children.
Now to read a bit more into the question your therapist asked. I take the question to mean: "How do you feel about yourself inside?"
Are you a woman? Or do you see yourself as a transwoman.
I consider myself to be a woman with a transsexual condition. I've always been female, but with a birth defect. I have always been a woman and other women can have other afflictions, like diabetes. It makes them no less female to be diabetic. And I consider myself no less female because it took a long time to diagnose my condition.
I'm a bit long winded today, Maggie, I hope I haven't bored you.
-Sandy