1. She should not have blurted out a patient's name no matter what. I've been in this situation - the same therapist as a friend - and all three of us knew it - and we all knew that we all knew it and STILL the therapist never indicated knowing the friend when I mentioned his seeing her. Because she didn't have his permission, nor was there any reason to ask for it. That's the way it should be.

2. If she felt that it would be useful for her to know the name - she could just have said so and made a case in general terms ("I thought it might help me understand such and such blah blah"). She could have explained the general principles of confidentiality and how she thought it did or did not apply here. Pressuring you and trying to "analyze away" your understandable reluctance is a big red flat you're absolutely right. That, to me is the biggest red flag, because it's a real mind game.

3. I have had one psychiatrist refuse to take a family member of mine as a patient because he thought it would be complicated to deal with possible conflicts of interest. AFAIK there's no ethical guideline against treating friends or family - but I thought that was a very proactive, reasonable position. i.e. It *might* make it difficult to be honest about your feelings about a family member if you knew that you were talking to that person's psychiatrist and it might affect their relationship. You might just want to say nice innocuous things. And it sounds like you're experiencing just that internal conflict.