Answer

Both. Our nationhood is defined by its religion (“Our nation is a nation only through its Torahs” — Saadia Gaon). Therefore, someone who is ethnically Jewish—that is, whose mother is Jewish or who converted according to Jewish law—but whose culture is now not religious and who is not committed to the commandments, is in fact not Jewish. Of course ethnically yes, but essentially no. His belonging to the nation stems from the religious criterion, but on his own view he is not Jewish. In other words, secular Judaism (as distinct from a secular Jew) is an oxymoron.
But this question has no practical significance. After all, others will define Judaism in a technical way (speaks Hebrew, reads Meir Shalev, and serves in the army), and by their definition they would be right. So this is only a question of definition.