In another thread Richard has mentioned some geographical terms, collected here. I'm not sure it's entirely right. Here's my understanding; do others understand the terms differently?

Richard: it is common, but wrong, to refer to anyone who lives in the UK or Britain as "English". Correct: 'Britain' includes England and other lands.

Richard: The British Isles excludes Ireland, which is not British. The Geographical term is the British Islands. Incorrect: 'British Isles' includes Ireland, as well as Britain and other islands.²

Richard: In much the same way it is common, but wrong, to assume that anyone who lives in America lives in the USA. True as phrased, but potentially misleading. It is proper to use the term 'American' to refer to the entire continent, but 'American' can also properly be used to mean "USA only."³

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Per OED:
¹Britain: The proper name of the whole island containing England, Wales, and Scotland, with their dependencies; more fully called Great Britain
²British Isles: a geographical term for the islands comprising Great Britain and Ireland with all their offshore islands including the Isle of Man and the Channel Islands
³American: 1. a. Belonging to the continent of America. … 2. b. Belonging to the United States

A nice Venn diagram of the British Isles terminology at Wikipedia.

My memory served me wrongly about the British Isles, which does include Ireland but excludes the Channel Islands. However, the Irish do not always like to be considered as part of the British Isles and the Irish Government rarely use the term.

The geographical term "the British Islands", however, includes all the many islands of the archipelago. According to Wikpedia:

"...Since 1978, the term British Islands (as opposed to British Isles) has been used by the UK's governments and assemblies to define the United Kingdom, together with the Isle of Man and the Channel Islands...."