You're on the right track! The terms "first quarter moon" and "waxing crescent" refer to different phases of the moon, but they are often confused. Here's the breakdown:

Waxing Crescent:

* Appearance: A sliver of the moon, less than half illuminated, and growing larger each night.

* Illumination: Less than 50% illuminated.

* Position: Located between the New Moon and the First Quarter Moon.

First Quarter Moon:

* Appearance: Exactly half of the moon is illuminated, with a distinct, sharp, semicircular edge.

* Illumination: 50% illuminated.

* Position: Occurs about a week after the New Moon, and one quarter of the way through the lunar cycle.

In summary:

* Waxing Crescent is the early phase of the moon's growth, showing a thin crescent shape.

* First Quarter Moon marks a specific point in the lunar cycle where the moon is exactly half illuminated.

Think of it like this: The Waxing Crescent is like the "baby" phase, while the First Quarter Moon is like the "teenager" phase – still growing but with a more defined shape.