The diplomatically neutral term is "the island of Ireland" for the whole thing.

"These islands" refers to the British Isles, in the context of Anglo-Irish relations.

It's some fascinating wordplay, born out of the Good Friday Agreement, trying not to step on various people's buttons.

It's not really ambiguous. When talking about Ireland in an economic sense, we generally refer to the Republic of Ireland. Geographically you would say the island of Ireland if you were to include Northern Ireland.

Do you also avoid using the names of Korea, Georgia and China?