The woman does use my name all the time - I have no idea why. I know she is talking to me - I am the only other person in the room. I do not address email or letters to her by name- I don't use anything. I don't think I have ever used her first name when speaking to her - there has never been any reason to use her name (I would use her first name if I had a reason to use her name at all - I just never have had). Even if I talk about her to the two people who sort of know I see one, I call her that woman and do not use her name.
If I sign the letter - I only use my last name or, if very put out, I use my whole name with title.