Yes, it’s a myth. 100%. There’s only 2 ways of reading “Irish people weren’t considered white”. The first is that people didn’t consider them white in the consensus definition of the word: white skin/being of white stock/white-classed ethnicity - which Irish were considered to fit this (thus the claim is a myth).
The other way of interpreting this statement is that white has any definition other than the commonly accepted one. In this case, not only has it led to so much misinformation (people genuinely think Irish people weren’t considered white people), but in this particular definition of “white” it is racist and assumes white supremacy by definition by just putting anyone discriminated against in any way as “definitely not white”, an entirely politically motivated definition, and so should be entirely disregarded.
“Irish people weren’t considered white” is a myth, it’s completely wrong. If you are trying to say something like “Irish people received more discrimination than anglo-Saxons”, then just say that rather than appropriating the term white for political purposes.