For the Olympics you are incorrect but it varies.

Rugby - one team for the island

Soccer - one federation for each polity on the island, but Derry City plays in the League of Ireland (Republic of Ireland). Two international teams.

Cricket - one team for the island

Etc etc. There is some tiresome hand-wringing every time an athlete has decide on their allegiance.

Basically, Ireland is a political Mandelbrot set.

> Each island competes as one.

I don't think that's technically correct. Anyone in NI has Irish citizenship (who wants it), so any NI athlete can represent Ireland.

"Team GB" is the official team of the whole of the UK though, including Northern Ireland.

You are correct, the official name of the team is `Great Britain and Northern Ireland Olympic Team`

They don't at the Olympics.