As for the more general issue as to "Who is a Jew?", you have to agree that there are some cases where this is a thorny problem. Israel has had to define Jewishness for the purpose of immigration policy, and that legal definition of "Jew" in Israel has caused some controversy. Most notoriously, there have been young Russian immigrants to Israel admitted as "Jews" who have no immediate Jewish family background and who have caused a lot of trouble in Israel by perpetrating antisemitic incidents.
Israel doesnt rule on whos a Jew, those issues would be dealt with by the Rabbinate if there was a question. Israels law of return is based on persecution as a Jew, not ones status as a Jew. A Russian with a Jewish grandfather but a gentile mother who hasnt adopted another religion can immigrate, since a Jewish grandfather could be the basis for persecution. Under the Reich, whose standards Israel adopted. However he would not be considered a Jew. Unless he converted. Orthodox conversion only, which is the issue youre alluding to. But many Russian immigrants arent considered Jews by the Rabbinate.