I still maintain that "The Irish" don't claim to have composed these songs.

All three links that you have given Rob, credit fiddler's Green to John Connoly. The first link is to subpage on a UK website. Link 2 is to a website based in Austin Texas and in Link 3 Martin Dardis does say that he has gathered songs from around the world, especially from England, America and Germany which has a rich culture in folk songs"
With Fiddler's Green however it's easyto see how the non-Irish could assume it is Irish because:

John Connoly is an Irish name
The word Green in the title
The Dubliners sang it!

Or in the case of Dirty Old Town:

The Pogues sang it as well!

But as I said earlier these songs are now part of the Irish tradition ...