Translation:The President of the United States is in the White House.
Actually the 'na' introduces the defined genitive '(of) the States'. Uachtarán is made definite by the usage of the genetive 'of the state' to qualify it. 'na Stat' is used a bit like an adjective, defining what 'president' we are talking about. To say 'a President of the USA' you could therefore not use the genetive! But rather use a preposition like 'de' (=of)
Cant reply to your reply so replying here. True about Leabhar leabharlainne. So that brings us to a middle ground: the definite genetive makes the noun it defines definite, whether there is an 'an/na' or not, as with Rí Shasana.
So from a practicle point of view, it is right to point that the article in front of the genetive is a good indicator of the Uachtarán being definite. But technically the article defines the genetive and the defined genetive defines the noun. Technically.'na' does not define both, but indirectly it looks like it... When there is an article.
Thus why Shasana, defined genitive without article, follows the same rule as Uachtarán na SA... but is not picked up by your rule of thumb. Because Sasana does not need the article to be defined...
Read your last sentence: 'de... na...' The 'na' is still there and still playing the same role: to qualify SA (in the nominative this time). The difference is that 'na SA' nó longer behaves like an adjective qualifying a noun, but like a noun introduced by a preposition.
You could have Rí Shasana = the king OF England. The genitive makes Rí definite...
And, yes, the na is there. The difference is in the non-genitive phrase it only quantifies SA, in the genitive phrase it quantifies both, since Irish can't have a structure like An X an/na Y
But I think that has more to do with the semantics Sasana than the genitive itself. Leabhar leabharlainne, for example, wouldn't be 'the library book'; it'd be 'a library book'.
Because they are two different things in English. It may be "equivalent", but it is not "the same" or "translation alternatives".
By contrast, you could translate "to make" or "to do" in many ways, some of which can be validly interchanged in translation.
But each of the terms "American", "US", "United States", "United States of America", have their own equivalent, and we cannot expect the course creators to stick to every "equivalent". otherwise, what stops us from asking them to translate "America" by "Uncle Sam"...
(Incidentally, USA in Irish is SAM...)