I’ll take a cursory stab, but other references exist for the minutiae of how these things came to be.

Britain == United Kingdom

Great Britain == an island wholly within the UK

United Kingdom: a sovereign state (eg USA, Germany) composed of the constituent countries of England, Scotland, Wales, and Northern Ireland, plus a few overseas territories.

Briton: a British citizen, or someone tracing their ancestry to the UK constituent countries

The short answer for why a country (UK) can have countries (eg Scotland) in it is because the notion of sovereign states (the modern definition of countries) only came into existence in the 18th century or so.

Canada, formerly the Dominion of Canada, formed from the British North American holdings plus the French parts that the British bought (ie Quebec). Granted self governance in the 1860s, independence in the 1930s, and finally full “patriation” in the 1980s to remove all vestiges of the UK from Canadian laws. However, the independent Monarchy of Canada remains, and just happens to coincidentally follow the exact same selection rules as the British Monarch. So the King of Canada will be the same person as the King of the United Kingdom, even though the Government of Canada is no longer controlled by the Government of the United Kingdom. In both sovereign states, the King is a figurehead from where authority and governmental legitimacy emanates, and the current King of Canada continues the tradition since Queen Elizabeth II that the Monarch’s appointed Governor-General of Canada shall represent the Monarch in all Canadian matters, meaning the Monarch will not directly involve themselves. In fact, it’s frowned upon for politicians to involve the Monarch directly, except to kindly ask for a new Governor-General (see 1975 Australia).

Why still keep the Monarchy of Canada? The voters haven’t chosen otherwise, to pursue a republic or any other form of government. The same applies in Australia, although it’s slightly more complicated as each Australian State derives their state-level authority from the Australian Monarch, whereas Canadian provinces exist as a part of the singular Canadian confederation; each Australian State would have to sever their connection to the Monarch, or could do so piece-meal. Nevertheless, with the Monarch delegating power within each of his “realms” to the respective Governor-General, becoming a republic is a matter of passing bills in the parliamentary system. Barbados did exactly this in 2022, replacing Queen Elizabeth II with a republic. Countries formerly part of the British Empire (circa 1930) usually join The Commonwealth of Nations, which just means they’d like to keep trading with each other but without the whole colonialism thing.

The end of the British Empire was basically slow-going, since once some parts (eg Canada) received a measure of devolved powers (like having their own parliament), the door was swung open for other parts to demand the same. Post WW2, with the status of the UK greatly diminished in deference to the USA and USSR, overseas colonies became expensive and untenable. Plus, one of the basic tenets of the United Nations – put in place to avoid yet another world war – was the right to self determination. So the colonies had to go free.

Ironically, the Scottish Parliament was not restored until 1998, having legally stopped existing with the Acts of Union 1707 that merged England and Scotland into the United Kingdom, with a single parliament but otherwise separate institutions and laws. To this day, the English Parliament has not been restored, and thus England affairs are directed by the UK Parliament at large.

The British Isles includes both the Island of Great Britain (where Wales, Scotland, and England are) plus the Island of Ireland (where the Republic of Ireland, and Northern Ireland are), and a few smaller islands. So yes, every part of Great Britain is a part of the UK sovereign state, but the UK also includes the Isle of Mann and the upper part of the Island of Ireland, and some others.

In all circumstances, “Ireland” means the Republic of Ireland; that republic is almost never called “southern Ireland”. The whole history of British colonialism in Ireland is long, sordid, and full of misery, culminating with The Troubles of the 1960s and still causing concern post-Brexit due to the EU border basically dividing the Island of Ireland.

TL;DR: everything about British history is messy. Even their former colonies have messy history, despite some being at most a few hundred years old.