My understanding of how the islands are named.
England is in the main bulk of land but doesn't include Scotland or Wales. They are geographically attached to England and that whole lump of land is called Britain. Even if another country invaded and re-named the country, the lump of land is called Britain.
The British Isles includes Britain and the 'bits' around it, e.g. Portsmouth, The Isles of Scilly, Mann, Hebrides etc.
The United Kingdom includes Northern Island.
Ireland is the name for both the piece of land, and the nation which is Ireland. Some people refer to Ireland as the Republic of Ireland as a way to differentiate it from the whole piece of land, or to describe its constitution, or out of error.
There are other islands which come under the protection of the United Kingdom, but they are not part of the UK. They recognise Elizabeth II as their monarch. These include places such as the Channel Islands and Tristan D'Cunha.
Then there is the Commonwealth. Basically, these are places we appropriated and managed to populate to such an extent that white people became dominant and we destroyed the people already living there. Eventually, rather than face the embarrassment of losing another country, like we did the US, we allowed them to have their own parliament and autonomy, stopped making them pay taxes, and have given them rights to migrate to the UK in a slighter easier way than non-Commonwealth countries. Australia, Canada, and New Zealand are the best known Commonwealth countries, but I've just checked and there's 53 countries in the Commonwealth.
I know chuff-all about the traditions for names, though.