Are persons from the Isle of Man, the Isle of Wight, the Hebrides, the Orkney Islands, the Shetland Islands, the Scilly Isles, or the Channel Islands considered British? The archipelago is known as the British Isles, but, for some reason, the term British seem to only apply to persons from the largest island in the group. Is this merely the strangeness of Irish nationalism damaging the rationality of the lexicon?
Actually, the term 'British Isles' isn't popularlarly or commonly used in Ireland, nor does it make any geographical sense. Nothing to do with the 'strangeness' of Irish nationalism, it's a simple fact that Ireland isn't British. Even Northern Ireland, while part of the UK, isn't part of Great Britain - the United Kingdom is of Great Britain and Northern Ireland. Ergo, even NI isn't technically British, even if most of its citizens choose to have British nationality (which they're perfectly entitled to do, AFAIC).
All islands of the archipelago are British by technicality. Britain was not until recently a short form of Great Britain, but the collective name of the islands. They were originally (so far as known) named this (Brettaniai) by the Greek traveler Pythieas in 320 BCE, and the name carried forward through Roman times (Brittannia, later Britannia), and through several variants in French and English until its present (rather Latinesque) form. The name is believed to originate with a Celtic word something along the lines of Pretani (the indigenous Celts name for the people of the British Isles).
The tradition of naming the island containing England, Scotland, and Wales Great Britain is relatively recent, and itself only denotes that the island is the largest of the island group. The ancient (as far back as Roman times) names of the major islands were Albion (Great Britain) and Hibernia (Ireland).
And as has been pointed out elsewhere, many people in Scotland, Wales and the Isle of Man don't regard themselves as British. But I really don't want to hijack the thread any further. Suffice it to say, that while regarding myself as Irish, I have no disdain or dislike for English, Scottish, or Welsh people or for Northern Unionists. Generalisations such as the ones you've made aren't helpful.
