Wittgenstein's question is a great one to ponder. It seems to me that the distinction between my raising my arm and my arm's rising is that raising my arm is intentional. That is, I raise my arm only if I want to, or decide to, or intend to raise it. But my arm would rise (without my raising it) if I fell into some water that pushed it up. It would also rise (without my raising it) if someone else--say, a hypnotist, or a strong person who grabbed it--intended for it to rise. In such a case, there would be an intentional rising of my arm, but I would not have raised it. So, the difference between my raising my arm and my arm's going up is that the first case--but not in the second case--requires some mental phenomenon on my part. (I am not attributing this answer to Wittgenstein; rather, this answer seems reasonable to me.) I'm not sure what you mean by a "biconditional link." You can't raise your arm without its going up, but it can go up without your raising it.

Existentialism is not really a single movement as it is a general outlook. Thinkers as diverse as Pascal, Dostoyevsky, Nietzsche, and Sartre have been called 'existentialists.' Only the latter two were atheists. (Pascal is famous for his proof for the existence of God--Pascal's Wager.) Nietzsche and Dostoyevsky agreed on the proposition, "If God is dead, then everything is permitted." But whereas Nietzsche found the prospect exhilarating, Dostoyevsky found it terrifying. Some existentialists are theists (e.g., Buber) and others are atheists (e.g., Nietzsche). I venture to say that of the existentialists who hold that there is no God, most would diesagree that everything is permitted. Existentialists (theists and atheists) are apt to hold that actions committed in "bad faith" are not "permitted". They are also apt to create or find meaning in life.