What is Corcaigh? Is it a name of place? The .teanglann.ie dictionary does not have this word.
There isn't a place called "Dublin City" (at least in Ireland - maybe one of the 13 or so Dublins in the US is officially called "Dublin City").
Dublin, like Cork, is both the name of a county and a city, so you might reasonably say "from Cork city to Dublin city" (lower-case "city"), but Baile Átha Cliath is just "Dublin".
Thanks for proving my point. Right there in the logo it says Comhairle Cathrach Bhaile Átha Cliath - "Dublin City Council", where Cathrach is the genitive of Cathair - in other words "The City Council of Dublin".
If Baile Átha Cliath meant "Dublin City", then "Dublin City Council" would just be Comhairle Bhaile Átha Cliath.
Cork City Council has a similar site - http://www.corkcity.ie/. Comhairle Cathrach Chorcaí - "Cork City Council", meaning "the City Council of Cork", not "the Council of Cork City".
Ring a Ring a Rosey As the light declines I'll remember just "Dublin" In the rare ould Times.