I had to mention this to my friend recently. She's in her late twenties. She used coloured a few years ago and I winced but didn't say anything, then when she used it again recently I had to mention it. It is hard, especially because for most people it is coming out of a place of genuine niceness - but on balance I think it would be better for me, as her friend, to mention it quietly, rather than someone judging her for something she didn't know she was doing wrong, iyswim.
Wrt to 'people of colour,' rather than 'coloured' my understanding is that it is a bit like 'person with a disability,' being better than 'disabled person' in that the 'person' is first and then the description/qualifier, rather than the disability/blackness being the most noticeable thing about them. It suggests that we are all one race of people, with different 'aspects' to us, skin colour, health issues, etc, rather than disabled people or coloured people being a completely different, separate category. Sorry if that's explained badly.
The other explanation is that people who have been discriminated against for whatever reason have the right to choose the description they prefer to use, and it doesn't have to make sense, or be approved, by anyone else. Which is also fair enough.