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Not to mention the fact that, as far as I am aware, the term used for the constituents parts of the UK is "country":

I don't think many people answer "British" to the question "What country are you from?".

Not many people would answer "British" to the question "What country are you from?", because "British" is not a place, but a nationality. ;)

If you asked me what I am, I might respond "British" - whereas if you asked where I'm from, I might respond "Britain".

Where are the isles of man, guernsey and jersey in that list of constituent "coutries"? That is, where do they fit within the kingdom?

None of them are part of the UK, and they are also outside the EU. Somewhere I have a Jersey Pound...