You are correct! The side of the moon that is always away from the earth is light or dark depending on whether it is pointed toward the sun. Thus, at new moon, the "dark side" is pointed toward the sun and actually is brightly lit up, whereas at full moon the same side is pointed away from the sun and is truly dark. To confirm this behavior, you should look at the illustrations under "Behavior of the Moon," "Phases," and "Does the Moon Rotate on its Axis."