As a white British person I have no experience of having to consider how my 'ethnicity' should be represented. Black, Asian, and Minority Ethnic - BAME - has been used to group people together. Of course, there are many strands to ethnicity and it's helpful to have a common and accepted phrase or word to describe a group. 

In the US, 'people of color' is a commonly used term that is often seen as disrespectful in the UK, due to the historic use of the word color. 68% of people currently classified as BAME would be happy with 'Ethnic Minority' as an umbrella term, so maybe it's time to move to what the clear majority prefer.