Yes, you are learning that living authentically is much more comfortable than going from one closet into another.
There is no embarrassment in being authentic. The nomenclature of the english language, however, has very few adjectives to adequately describe our situation. Actually none.
Yes, your daughters friend head may have spun around a bit, but he will get it or not.
Simply presenting yourself as her "parent" covers all the bases. And if someone questions the same sex arrangement, then you can go into details. Otherwise they can draw their own conclusions.
For me, I don't wear a sign around my neck that says "TRANSSEXUAL!!!". I simply live my life as it is. And if my background or experience is germane to the situation I go into it.
I think you did quite well.
I am reminded of the discussions about the emergence of the electronic synthesizer from decades past. Prominent in that discussion was the participation of Wendy Carlos. She had worked closely with Robert Moog, the developer of the Moog synthesizer, and whose architecture framed all other synthesizers since. Wendy used Moog's equipment in her creation of the most popular classical music record of all time "Switched on Bach" Though it was originally published under the name of Walter Carlos. During that period Wendy was also in the early stages of transition, it quite a difficult time for her.
In the discussion of that time, Bob Moog talked about the influence that Wendy had on his efforts and how the two of them worked together to design and modify Moog's equipment for her work.
He would say things like; "Well, Wendy would come to me and say can you make it do this..."
He didn't say "Well, Wendy (back when (s)he was Walter and while (s)he was starting to take hormones and experimenting with dressing in public and going through that awful time in the 70's when being transsexual was such a hard time for trying to transition), (s)he would come to me and say can you make it do this..."
The history of electronic music synthesis had nothing to do with a persons gender and simply not germane to the discussion.
Yes, don't hide, be authentic. I'm proud of you all! It sounds like you actually got a bit of a giggle out of it too.

-Sandy