The Nazi definition was designed to be easily confirmed through existing records. Germany had long been keeping records based on religious affiliation, so that is what they used to "weed out the Jews" regardless of the "religion" of the person to be murdered. Many of those killed, including my family, did not identify at all as Jews. The Nazis explicitly referred to Jews as a "race" which needed to be exterminated, and they were efficient at doing that.

See https://aninjusticemag.com/my-response-to-whoopi-goldberg-and-her-defenders-f620a4408cb8

When the Nazis conquered other countries, they relied on the racism of the people there to identify the Jews. In my Grandmother's home town in Poland, her Polish neighbors gleefully helped the Nazis identify the Jews in the community. First, they identified the leaders - including my Grandmother's father and brother. They were shot on the spot. Then her former neighbors identified the rest of the "Jews." They were forced to dig their own mass grave. They were all thrown into the hole they dug, where oil was poured over them, and they were all burned alive.