I've been checking Ireland extensively. Ireland's name is Ireland and it is referred to as such in treaties such as the Accession treaty to the EU which I just read part of today. This treaty see
http://www.bmdf.co.uk/ukaccessiontreaty.pdf It refers to "Kingdom of Denmark, Ireland and the United Kingdom of Great Britain and Northern Ireland" Note that Denmark is 'Kingdom of' but Ireland is not 'Republic of'
There is a historical reason for Ireland's name. When Ireland first seceded form the UK it was not a Republic. The Queen remained the head of state in the same manor as for Canada and Australia. Also, the state was called the 'Irish Free State'. In 1937 the name changed to simply 'Ireland' but it was still not a republic. I can imagine the name was contentious at the time, as it still is a bit in the UK now, since it implies that the Republic and the island should be coterminous. It did not become a republic until 1948. However, the change in form of government did not include a change in the name. It remained 'Ireland'.
Both names, Ireland and Republic of Ireland carry strong political connotations regarding the independence of Ireland and the ownership of Norther Ireland. Within the UK, Ireland is often referred to as 'The Republic' and northern Ireland referred to as 'Ireland' in order to exert a political opinion in favour of Norther Ireland Unionists. Such usage has diminished since my youth as 'The Troubles' have largely calmed down and the Stormont Assembly setup. (Though it's currently suspended.)
The use of names of counties frequently has a bias. Often, due to language barriers, we are not aware of the bias. So to avoid being accused of bias, we must choose names very very carefully.