Why would the speed of the shadow across the Moon Earth be traveling Eastwards?
Because that's the direction the Moon is moving, and the shadow will go this way too. Look down upon the Earth/Moon system, the Moon is going anti-clockwise around the Earth, so, if you got between Sun and Earth, with your head above the plane of lunar orbit, your feet below, the Moon would move left to right across your field of view.
Same as if you had a clock that was running backward. Tilt the face back to place you below the 6, perfectly in line with the face, and the ends of the hands will, as they pass the 6, move from your left (9) to right (3)
If the Earth were not there, and you had a flat circle of paper where Earth should be, the shadow would go from left (West) to right (East), i.e. Eastward.
No?