I don't think that a completely secular Jew would be Jewish at all. What does "Jewish" mean? Does it mean coming from a Jewish background? But we don't say that someone is a Christian simply for coming from a Christian background even if they are not a believer themselves. I agree that it is hard to be clear about this sort of thing because religion and culture are so closely linked, and this is particularly true of Judaism, which, more than perhaps most religions, is especially associated with various festivals, cultural practices, and so on, and has far less of an emphasis on doctrine. Plus it does not encourage conversion of outsiders, so belief in the religion is very closely associated with coming from that culture. All of this is a major factor behind the controversy over Beta Israel (better not say "Falasha", as this is a pejorative term). However, I do think that it makes little sense to describe someone who does not believe in Judaism or do any of the traditional cultural stuff as Jewish, since such a person is indistinguishable from everyone else. Similarly, if someone converts to Judaism (which normally happens only when they marry one, but it's still at least theoretically possible to do it even otherwise), then they become Jewish, don't they? In which case, Jewishness is a matter of what you believe (partly) and what you do (mostly). Someone who believes and does the relevant stuff is Jewish, and someone who doesn't isn't. I don't really see what other criteria you could have.