Looking back, you seem to think the contradiction is that Dedekind's axiom says there are two classes, but then introduces a point which you believe is of neither class. But his axiom does not say that it is of neither class. The axiom, as you've stated it, is actually a little ambiguous. It refers to it as the point that "produces the division."
Anyways, if you are going to divide time then in your example, you could either divide it into one class of points that are strictly before the present, and one class that is before or during the present OR divide it into one class of points that are before and including the present, and one that is strictly after the present. Do you see how these would be complete divisions? If we took two classes, one that was strictly before and one that was strictly after, we wouldn't have completely divided time. Now, regardless of which of the above two ways you divide time, the present serves as the one and only dividing point. And depending on how you do this, the dividing point is either in the earlier class or the later class, but it is going to be in one of them, it's not in a third class. So what's the problem?