Yup! You're spot on in fact. I actually have an article all about plastic paddies: https://meron-cruithne.medium.com/i-am-fed-up-with-plastic-paddies-and-plastic-every-nationality-151832f036b4

There are Irish people of all racial backgrounds. Culturally, Irish means "born and raised" in Ireland (or raised in Ireland, in many cases). Irish-Americans definitely do not count - unless and until they move here and actually stay for a long while.