The first Irish were pre-Celtic, yes (the Tuatha de Danann, who have been surmised by various sources to be the Israelite Tribe of Dan; their earlier name “Tuatha Dé” actually means “Tribe of God”, which seems to link to that theory). That’s been known for ages. That link you gave me is indeed about the early Irish.
The Iberian Irish weren’t Celts.
Dublin and more were Scandinavian towns. Not Celts.
The Normans weren’t Celts.
The Celts were all over Europe. From Marseille to Latvia. So we are all Celts.