And yes, I too find the term 'mulatto' offensive, I didn't ever say it was acceptable.
Bugsy I wouldn't accuse someone of racism for calling my DD 'black' because at the moment, as has been outlined, it is not considered offensive by everyone and is not consciously used to be offensive. I think most people are more rational than to chuck accusations around when the intent to be malicious is not there. What I would do is tell the person point blankly that I do not approve of my DD being described as 'black'.
I have formed my opinion based on the way African/Caribbean people have been treated for many generations. When people have been described as 'black' in the past, it has not been with positive connotations, and it has been to create this picture of a large homogeneous mass, thus denying a 'black man' his individuality. 'Black men' were the bad element of society and this idea was applied to all 'black men'. The term 'play the white man' is born of this. So being called a 'black man' was not just a way to describe someone's heritage. It was an insult because when many people would call someone a 'black man', it was (and still is) loaded with stereotyping and misrepresentation of character.
You see this in the news all the time. When a 'white man' commits a crime, he is rarely described as 'the white male' or 'Englishman' as an example. But when a crime is committed by a person of African/Caribbean heritage, the persons skin colour or heritage is more likely to be highlighted in the article. This is because being titled a 'black man' still means you are branded as having certain inherent characteristics.
To be clear, if this wasn't the case, I would have no problem with people being identified as a 'black man' or 'black woman'. But I feel the description is too loaded and has been used to demean and dehumanise.