Didn't see this until now. The King list you provide for the Irish kings is the typical list taken from Foras Feasa ar Eirinn (Foundation for a knowledge of Ireland), Annála na gCeithre Máistrí (Annals of the four masters) and the other Annals.
However the list is almost entirely fictional, most of it was written and rewritten in an attempt to match Celtic beliefs and Christian beliefs and was then further distorted at various points in history to match the political aims of various families. The list and the surrounding origin myths (written in the form of prose poetry) settled into a canonical form in the late 11th century in a work known as Leabhar Gabhála na hÉireann (Book of Invasions of (the) Ireland).
The main point is that the book projects an 10/11th century phenomena (the high king) and attempts to project back into earlier history when in truth the title simply didn't exist.