@dorzak, I think the lack of formatting (headers etc) on that page leads to the confusion. Right below the section you quoted the following appears:

Entry into Ireland from outside of the EU
Dublin Airport is the only entry point permissible for pets entering Ireland from non-EU countries. These animals must travel as manifested freight (cargo) on an approved airline or with an approved cargo company.

Yes, but the list they link to are all countries outside of the EU.