Wow, "trivializing her profession," huh? To me, it's totally understandable that you would be reluctant to identify with a psychiatric diagnosis (whether it fits or not) and that you would tell your T essentially that it's easier for her to label you than it is for you to live with that label. (Again, the issue of how apt the label is actually immaterial.)
She might have said something like "Oh, you feel it's a hard thing to live with?" Or "Hmm... So you think maybe I've been hasty or cavalier, I wonder if there's information you think I'm missing?" Or just accept that you prefer not to have a diagnosis. In the end, especially if she's not a psychiatrist, how much does the diagnosis matter? It's your history and how it's affecting you now that matter, right?
So whether it's PTSD or sub-clinical PTSD or some other type of anxiety does it really matter what you call it? You work through it more or less the same way.
I don't feel reassured by her professional pride at all here. I feel alarmed by her insecurity. When she feels insulted she should be able to sit with that feeling, observe it, breathe and process it later. She should not be reacting to you from a place of hurt pride. This process is about YOUR feelings, that's why you pay her instead of chatting with a friend.