By all legal stipulations, there’s no indication that Irish people were not always considered white. There is not a single instance where Irish people were denied from “white” accommodations. They were always considered to be of “white racial stock”.

They received discrimination in other ways, including for Irish ancestry, but not for being white. Do you understand how those are different? The article DID mention “Irish need not apply” signs, but that’s on the basis of ethnicity, not race.

Different groups had different levels of acceptance even today. The vast majority of Americans consider Jews to be white, but there are still people who dislike Jews. Jews receiving discrimination doesn’t make them any less white.

Even your linked article has 0 examples of Irish people not being considered white. You seem to be under the impression that being white by definition can’t receive racial discrimination, and if they do then they can’t be considered white by definition. That’s a ridiculous definition, it’s confusing, and black/Asian/etc has no kind of definition so this definition is clearly just politically motivated - it overloads a commonly understood term to smuggle in the idea that white supremacy is just true by definition and whites people never experience racism by definition. Both of which defy observed reality, making this an entirely nonsense definition.

The only valid definition of white is the commonly accepted one: of the white racial group as decided by a combination of factors including your lineage, how you are perceived, and so on. This is the same definition for any race, and under this definition: Irish people have always been considered white.