The act which made "Ireland" independent was called "The Republic of Ireland Act", and reads as follows:
"... the description of the State shall be the Republic of Ireland..."
The constitution [0] reads "The name of the State is Éire, or, in the English language, Ireland", and that act can't override it. I suppose it's splitting hairs to distinguish between the "name of the State" and the "description of the State" and either can reasonably be seen as what the country calls itself.
> Notably, the Act did not change the official name of the state. It merely provided the description for the State. The Constitution of Ireland provides that Éire (or Ireland in English) is the official name of the State and, if the Act had purported to change the name, it would have been unconstitutional as it was not a constitutional amendment. The distinction between a description and a name has sometimes caused confusion. The Taoiseach, John A. Costello, who introduced the Republic of Ireland Bill in the Oireachtas, explained the difference in the following way:
> If I say that my name is Costello and that my description is that of senior counsel, I think that will be clear to anybody who wants to know. If the Senator [Helena Concannon] will look at Article 4 of the Constitution she will find that the name of the State is Éire. Section 2 of this Bill declares that "this State shall be described as the Republic of Ireland." Its name in Irish is Éire and in the English language, Ireland. Its description in the English language is "the Republic of Ireland."
The constitution specifies that the name of the country is Éire in Irish or Ireland in English, and that name is used in all official functions and diplomatic relations. The act is neither here nor there. The UK referred to Ireland as the Republic of Ireland until 2000, but it was unusual in doing so.
Nowadays, "Republic of Ireland" is usually only used where disambiguation from Northern Ireland is necessary.