Congratulations! you have noticed the moon's Relative Translational motion
I'm pretty sure that this displacement was to the east, or nearly, right? the Moon has an average relative motion of 13 degrees per day to the east.
Is called "Relative" because the Moon's eastward motion is much slower than the sky's westward motion, so though moving to the east from day to day, it still has a net motion toward the west each day. this means that it still rises in the east and sets in the west like the stars, but a little later each day.