I'm still not sure about this whole concept of 'atheist Jew'. To me, a Jew is someone who practices the Jewish faith, AKA Judaism. Someone of 'Jewish' descent would simply be Israeli.
So a follower of Judaism = Jew
A 'Jew' by blood or genetics = Israeli
I believe the term 'Israeli' fits because that is the country which blood/genetic Jews have claimed as their land. If being Jewish isn't simply about the religion, then what would the term be for someone who is religiously Jewish but not blood/genetically?
Fallen_Horse, that's ok a lot of people do not understand. Judaism is a faith, it's also a faith tradition and a peoplehood. One could say that Christianity came along and created a religion but before that there were groups of people and one group were Jews, hebrews, Israelites what ever you want to call them. My religious tradition says that Judaism is a people and a religious tradition.
Today the term Israeli is someone that is a citizen of the state of Israel. There are non Jews that are Israeli and have citizenship in the state of Israel.
There really is no "real" "bloodline", Hitler, the King and Queen of spain in 1492 and everyone in between claimed that Judaism was a blood line, a "race" but Jews really aren't. Yes you can technically be born Jewish because of your Jewish mother and today by having a Jewish father, but one could also convert into the tradition, which millions of people have done over the centuries.
As far as your question about religiously Jewish but not "genetically" Jewish. That would depend if they converted or not. If they did not convert my tradition basically says that person has a Jewish Identity. A Jewish identity refers to how an individual sees himself/herself, using this example a person might call themselves a Jew and even see themselves as a Jew. Status is how the rest of the Jewish community sees someone or how a community sees its member, or how the rest of the world views that person. In order to have Jewish status, and that person does not have a Jewish parent one has to go through a process and that process is conversion. I have met people that have taken on a Jewish identity but not a status.
I hope this helped