You might be wondering why Baile Átha Cliath bears no resemblance to Dublin.
The city of Dublin historically had two main settlements: The Viking settlement was known as Dyflin, taken from the Irish Dubh Linn ("Black Pool"), and the Irish settlement further up river was called Áth Cliath ("Ford of Hurdles").
The Viking settlement later became Anglicised to Dublin, but Irish speakers continued to call the city Áth Cliath (and still do to this day). Its full name Baile Átha Cliath means "Town of the Ford of Hurdles".
Bláth Cliath is a nice contraction, kindof like "flower of the hurdles". However, I'm not sure if this contraction strictly confirms with An Caighdeán. I dont know if Duolingo would accept "Dubhlinn" as a correct translation, it's commonly used too. No Idea where the officialdom of An Caighdeán stands on that one.
It's very uncommon to refer to Dublin as 'Dublin City', not just because the county no longer exists, or because city has swallowed a massive chunk of what was County Dublin, but because when people think of 'Dublin', they think of the city, not the county. So unqualified, when you say 'Dublin', you're talking about the city.
It's as unnatural to tack 'city' onto the end of 'Dublin' as it would be for London, Paris, Berlin, Brussels, Amsterdam, Rome, or any other major city other than the odd exception like New York City and Quebec City. But then it's because those cities share a name with a significantly larger area. The cities of Cork, Galway, Limerick and Waterford often have 'city' tacked on the end for this very reason: there's a need to distinguish them from their surrounding counties. Northern Ireland has two major cities: Derry and Belfast. Derry if often referred to as 'Derry City' to distinguish it from the surrounding (now abolished for administrative purposes) county, whereas Belfast rarely if ever is called 'Belfast City' because there's no need.
You'd never tack on 'town' to something the size of Dublin unless you were referring to the city in some bygone day when it was much smaller. You'll here that kind of thing in folk music and poetry, or in sentimental prose, but not outside of that. Dublin is much too big to be referred to as a 'town'.
That's specifically about the city centre, and isn't specific to Dublin. 'Into town' basically means that you're visiting the city centre just about everywhere in the country.
There's a difference between something like 'in(to) town', and either qualifying a city with 'town' as in 'Dublin Town' or referring to it as 'a town'.
I think you're mixing up the fact that Baile Átha Cliath has nothing to do with the modern name of Dublin.... read the above comments and you'll see why..... also, the point was being made that when you say "Dublin" it ALWAYS means the city, where as if you were to differentiate you would have to say Dublin County... which rarely happens.
You can pronounce Baile as bolya. Átha as aw-ha and Cliath as clee-a, but if you're actually talking with a fluent Irish speaker about Dublin, don't be surprised to hear "blaw clee-u".
The very first words in this TG4 documentary are I gcathair Baile Átha Cliath inniu
(That clip is no longer available - here's another clip where you can hear Baile Átha Cliath in the first few seconds - https://www.youtube.com/watch?v=v3-L00rOS-g)
Both pronunciations are correct.
Our first time in Dublin, I asked one of the locals (a guy who worked in a car hire/rental place) how to pronounce it. He said it several times, fairly slowly, for me. It sounded like “Bah lee aha cleeve” to my semi-deaf Texan ears.
I never did learn to say it right, no matter how much I tried!