Right. I mentioned Eastern Europeans as the predominant Ashkenazi group because currently they tend to outnumber other groups of Ashkenazi Jewish Europeans, but yes, the gene is also found among Western and Central Europeans as well.
I think part of the reason "Jewish identity" confuses, is that technically a person can be genetically of the Ashkenazim of Sephardim, but not necessarily practice Judaism. Then the question becomes: is that person Jewish because they have a gene connected to the historical practice of judaism, or are they not Jewish because they don't believe in/practice Judaism. ... And then you have silly rules that confuse everything further, like, "Well, you aren't Jewish unless your mother is Jewish". Or the equally silly, "You're only Jewish if your father is Jewish". That of course, makes no sense in the context of people who carry a so-called Jewish gene.
So in some ways genetic facts do get conflated with religious beliefs/practices or the lack thereof.