You already have your answer. You've received four accurate replies from four well-informed people, none of them either English or British, which is nice to see.
I'd just like to add something from within, so to speak. One interesting issue that came up in last year's referendum about Scotland is that 'British' is largely a political distinction. It's our passport, our nationality, the Union Flag under which we sit, publicly, on the international stage.
By contrast, being Scottish, Welsh or English ( or Northern Irish, if we're also talking about the UK) is more of a cultural issue. especially if you happen to be Scottish, Welsh or N. Irish. It's about ethnicity, history, values, attitudes, and way of life, what you look like and how you speak (language, dialect and accent). Scottish and Welsh people are very conscious, and rightly proud, of coming from a culture which different from that in England.
By contrast, 'Englishness' is very hard to define, and even harder to separate from Britishness. In some ways, being English can feel like a default state, for the 50 million or so of us who are British, but not Scottish, Welsh or Irish.