She didn't say she didn't like her behaviour, that would be different. She said she didn't like HER. I'm 26, I feel so sad for the ops daughter. She must feel so alone, and from her POV her mother has her perfect little nuclear family whilst she is the black sheep nobody likes. Thank goodness for her boyfriend. At 16 I said some horrible things to my parents, what 16 year old hasn't? At 16, her thought processes were not the same as yours. I agree entirely with the poster who said that in saying how great this other Mum was, she was noting affection and unconditional love, not "oh she's so cool she smokes and drinks" etc. in fact, by arguing those shallow points with her at 16 your daughter will have felt so much more misunderstood and alone than before your argument, she ran away from the hurt.
At 16, this girl left home yet worked hard to complete her a levels and get herself to masters level, yet the OP is taking credit for this, in some way taking away her daughters achievements. She might not have vocalised it directly, but her daughter will have picked so much up through her body language and actions.
We don't know the full story but her relationship with her father and the split might be extremely relevant here.
OP she is an adult now but that is not to say she isn't emotionally damaged. If you read through the stately homes thread you can see just how far adolescence affects women for life. I am not saying that you are a bad parent at all, as it is clear from your act of posting that you do show pride and concern for your daughter, but your posts do display some narcissistic signs. She can't just "get over" 10 years of feeling rejected as you'd like her to in your OP, she needs a lot of help and support, and so do you. Personally I would not write a letter, its cowardly and she will see it as another way of stepping back from responsibility (as with delivering the clothes etc).
She asked you to speak to her like an adult. Couldn't you try to do that, and listen to her without interrupting or fighting back, no matter how painful the things she says are? It would be an incredibly difficult thing to do, but one which a counsellor, as moderator, could make much easier.