Judaism is a religion, not a race. Someone isn't half Catholic or half Protestant, so they aren't half Jewish. Wesley Clarke by virtue of the fact that his mother was not Jewish, is not Jewish, per Jewish law (Halachah).
Again, technically you are correct. However, you are living in a dream world if you think that anyone who has a parent that is Jewish, is not considered 1/2 Jewish, whether you like it or not. That's just, realistically the way it is. Regardless of whether you consider this to be purely a religious consideration rather than a secular one. I don't mean say this with any negative connotation, just how the majority of the world considers it.
People don't speak of secular Jews as not being Jewish because they don't practice the Jewish religion or only their mothers are Jewish. They are still considered Jews, even though secular. I don't think there's much point in arguing about what I think you know is obvious. It's not a case of asking you to like this definition of Jew, it's just asking you to acknowledge that this is the way most people think, which is most consider Jewish like they would consider Arab, or Asian, or some other enthnic group. By usage, the term Jewish is looked at not merely in religious terms.
"Someone isn't half Catholic or half Protestant"
LOL! I am. My grandfather on my mother's side was Catholic; his wife was Lutheran. My father was Catholic, my mother was Lutheran. I was raised Lutheran, but became Catholic when I married a Catholic. That marriage was annulled. All my life I've lived in a religiously split family situation. Personally, I like both religions. They each have their good points and their bad ones.