Why does the Sobel operator, as the Wikipedia article also concludes, smooth the image out (with a triangle filter) perpendicular to the derivative direction, before it applies the central difference operator? What is the purpose of that?
Yes, that might much likely be it. My thought was that it would have something to do with the rotational symmetry of the operator, which is mentioned very shortly in the Wikipedia article (but not explained). I have to figure out what that is though.