Translation:Computers are available in the Internet café.
This is how I see it. Could be grammatically incorrect...
I gCaifé an Idirlín, would be 'in THE café of THE Internet'.
Sa chaifé Idirlin, is more like 'in the caifé of (AN) internet = in the Internet café.
This is how a noun is 'used as an adjective': it qualifies the other noun without defining it. (Second sentence) The caifé you recognize by an Internet connection... = THE café wirh Internet = THE Internet café.
In the first sentence, the 'an' of Idirlín is a defined article, making Idirlín defined too, thus defining caifé (which in turn no longer needs the 'AN', thus the use of 'i' instead of 'san'.). It becomes THE (defined) café of THE (defined) Internet. THE café people on THE Internet patronize... It could be the name of the jount even!
If you wanted 'A' café of 'THE' Internet, then you would not use the genetive: sa cháife den Idirlíon... (not qualifying, not defining, just connecting: so caifé needs AN to be defined)
You would say "THE internet" when it is the main word, on its own, as a subject (or object):
Tá an t-Idirlíon ag obair: the Internet is working.
Úsáideann sé an t-Idirlíon: he uses the Internet.
Tá an t-eolas ar fáil ar an Idirlíon: The information is on the Internet.
In those cases you use "an" ("THE").
But in the sentence at play in this exercise, the Internet is qualifying another word, so the dynamic is different.
When you think about it in English, the "THE" in "THE Internet café" is about "café". It could be "THE red café", "THE French café", "THE café-en-Seine".
So "úsáideoir idirlín" = "(an) internet user".
I think that, as is often the case, it's not our uses of the Irish language that's the differentiator here. I think that our English dialects (or at least our personal uses) differ. That's perfectly fine! :) I'd always use "the internet" in preference to "internet", even as an adjective or adjectival phrase.
Usually the genitive is used where you would use "of" in English. You clearly picked up on the case of the possessive (you asked about "Cafe's Internet") but you need to unroll things a bit for the possessive - instead of "a man's hat", expand it to "a hat of a man" - hata fir
Note that it is the "possessor" ("man's") that is put in the genitive (fir) so in the case of caifé Idirlín it would be "café of Internet", but obviously we wouldn't use a possessive structure ("'s") for this in English, we recognize that using Internet as an adjective to describe the café makes more sense - a "café of Internet" is an "internet café".
Then don't bother learning the genitive of every word for now - it's not worth the effort at this stage of the game. It's much more important to learn when to use the genitive - after all, it's a waste of effort to learn the genitive if you forget to use it in the places that the genitive is needed. You could be penalized in English to Irish translations that require the genitive, but if you recognize the sentence needed the genitive, you can allow yourself to look it up before you submit your answer, whereas knowing the correct genitive won't be any help if you don't realize that it is needed and enter the nominative. Similarly with Irish to English translations like hata fir recognizing that it's a genitive construction will tell you to pay attention to whether fir is singular or plural.
By the time you are comfortable at recognizing where the genitive is needed, you'll be experienced enough to start recognizing the patterns that are used in forming the genitive.
Ha Ha, Other than possessives I have no idea where the genitive is actually needed. It is always a surprise to me when it shows up (and then usually I read in the notes and someone tells someone else that it actually is the genitive, otherwise I would never know, what it even was). What I really need is an Irish language learning book that breaks it down more. If I had a book that made you practice using the genitive when you have a noun used as an adjective...THEN I might remember it but most of my books either don't get into the subjects enough to get to this level or they go through everything too quickly with not enough exercises....I have not given up though, ordered a new book today :)