I have no deep insights into British mentality. But I think in large parts it would relate to the context and the audience.
Were I in India, and someone asked where I was from, I would say "America". Were I in New York, I would say "Illinois". I can reasonably expect a New Yorker to appreciate the difference between Illinois and Virginia; a Bihaari, not so much. Similarly, my observation has been that someone from England will represent themselves as British - or possibly English, Scottish, or Welsh - to someone who is not British, but will go into more detail in contexts in which it makes sense.
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Were I in India, and someone asked where I was from, I would say "America". Were I in New York, I would say "Illinois". I can reasonably expect a New Yorker to appreciate the difference between Illinois and Virginia; a Bihaari, not so much.
Similarly, my observation has been that someone from England will represent themselves as British - or possibly English, Scottish, or Welsh - to someone who is not British, but will go into more detail in contexts in which it makes sense.