Is it context sensitive, for example, it will change if they are abroad or not; Or if they mean to evoke culture versus national origin?
If I am speaking to anyone from any other country in the world then I am British.
If I am speaking to an Englishman then I am Scottish.
If I meet another Scotsman in Scotland then we are British but if I meet another Scotsman in England then we are both from up the road. If I meet another Scotsman abroad then we are Scottish.
If it is sport then I am Scottish unless that sport is Team GB then I am British although Team GB victory will rely on a Scottish athlete, not a British athlete.
If any other nation is playing England at sport then I am that nation regardless. I would rather support North Korea as they flog their players to victory than see a single goal scored by England.
It should be noted that the 4 nations of United Kingdom of Great Britain and Northern Ireland have different cultures, accents, dialects, diets and religious makeups although all are primarily Christian. However identifying as Christian is loosely defined in the UK and not really important unless you want to fight with other Christians about football or live in Ireland.
Wider Identity Crisis
The Best Bit
The Bank of England was invented by a Scotsman and the religious argument in Northern Ireland is really about the Monarchy. Cornwall don't actually like anyone in the UK and believe they are special and Wales and Scotland are united in hatred of the English. England believes they pay for everyone and Scottish MP's can vote on English matters but English MPs cannot vote on Scottish matters.
It's all terribly complex but as a Brit you understand it without being told.
That's without even getting started on class...The Scottish are working class and the English are middle class and upper class. Although Northern England is working class and hates Southern England because they are upper class. So Northern England feels more cultural affinity to Scotland. Everyone hates London (the banking class) but for different reasons. Old money sneers at New money and new money sneers at Middle Class. London sneers at the UK. Modern economic conditions have actually made the Middle Class the new Working Class and the Working Class (blue collar) are actually the new middle class because a plumber is worth more than a University graduate...
As a Scotsman I think I can answer this as definitively context sensitive.
A current event or time served with the armed forces re-instills a British identity. .
But classically in sport achievement, when an Englishman loses he becomes a Brit.
And in winning, he remains English, assuming he's English. If he is you'll be sure to know it. (If he were Scottish, he would become British!)
I think that recognizing oneself as British (with any profundity) can be related heavily to a subject's own education in history and modern studies and awareness of the world stage, and whether or not you received free milk at school as a child.
To be born in the British Isles or to hold a British crown passport creates a British identity which is used interchangeably but typically held in reserve.
Domestically 'Britishness' is often secondary to the overwhelming distinctively separate regional/national identities that can be found in the UK. The likelihood is to identify oneself as English, or Scottish, or Welsh, or N.Irish(.) by default.
As someone born in England I find this question quite difficult. I usually try to phrase it as "I'm from the UK", because both "British" and "English" have strong negative connotations.
English has become associated with nationalists, white supremacists and others on the far right. It's also become associated with a narrow, protectionist view of the world (the "little Englander").
British has become associated with our poor international political reputation, due to Brexit and low quality politicians representing us. We also have a somewhat bad reputation on issues like mass surveillance, human rights and xenophobia.
As such, I try to avoid both and say something like "I live in the UK", which is pretty much the minimum possible association it's possible to express.