The issue with "coloured" is pretty simple but seems to confuse many for some reason.
We're all coloured but the term is used to generalise and group people that are non-"white". That's a massively diverse group whether it's based on nationality, race, skin colour, culture, etc. It doesn't actually tell you anything accept that they're non-white.
Turning it around how many caucasions would be happy to be described as, for example, non-yellow instead of white
Historically the phrase "coloured" has been applied to those of African/Afro-Caribbean origin, hence that particular demographic uses it today as a stick to beat the white population with. You don't hear those of arabic, latin, asian or oriental origin getting all huffy when the phrase is used.
Equally, using the phrase "white" to describe a certain section of society is just as generalised in it's own way as using "coloured" , but no one who fits that demographic gets the arse when it's used. But people could... I
could, for example, find it offensive to be described as just being "white" because I'm actually Northern European Caucasian and don't want to be grouped in the same category as Southern Hemisphere Australasian Caucasian types. Whenever I have to fill in a form that asks what my ethnicity is I always choose "Other" and put White, Scottish because I'm sick to death of seeing a dozen or more options for every other non "white" ethnic group....but just one option for those of us who just happen to be "white and British".
The main reason that the word "coloured" offends some these days is because they associate it with so-called white imperialism and they're too stupid, militant or racist themselves to differentiate between history and a generalised description of someone's skin colour.