All the stars in the sky are also rising in the east and setting in the west. Our own sun is rising in the east and setting in the west. The earth's spin causes stuff in space to appear to go in the same direction.

Stars in general appear to move across the sky (east -> west) faster than the moon does. Does this mean the stars are actually moving faster than the moon? Of course not. The sky is just a stationary backdrop, and the moon is moving along the same direction as Earth's rotation.

In the eclipse situation, the actual speed of the Moon matters, not the fact that its orbit is so long that it orbits the Earth about once a month.

The apparent relative motion with respect to the Sun is what matters. Of course, both bodies are rising in the East and setting in the West. But at the moment of the eclipse, the Moon is a little slower.