> What agenda? I do not know enough about the history to understand the implications

Traditionally, you mostly see weird names for Ireland in British media with an anti-Irish viewpoint (you saw a lot of this from the right-wing tabloids during Brexit, in particular). In particular ‘Eire’ and ‘the Irish Republic’ are red flags; ‘Eire’ has never been correct to use in English, and ‘the Irish Republic’ was never _really_ the name of the country at all. At best it comes across as ignorant (a bit like Americans calling the UK ‘England’) but in practice it tends to only be used by a certain type of publication.

(In particularly extreme cases you sometimes see ‘the Irish Free State’, presumably from journalists who are over a century old.)

> How does one unambiguously refer to the country rather than the island without saying “republic”?

Ah, well, that’s the trouble, isn’t it? In cases where it could be ambiguous, Irish media tends to say ‘the state’, but that doesn’t really work elsewhere. In practice where disambiguation is required “the Republic of Ireland” is a reasonable neutral term.