half-moon vs phase of the moon

half-moon and phase of the moon both are nouns.

Word	Noun	Adjective	Verb	Adverb
half-moon	Yes	No	No	No
phase of the moon	Yes	No	No	No

As nouns, phase of the moon is a hypernym of half-moon; that is, phase of the moon is a word with a broader meaning than half-moon:

half-moon: the time at which the Moon is at first or last quarter when half its face is illuminated
phase of the moon: a time when the Moon presents a particular recurring appearance

half-moon (noun)	phase of the moon (noun)
the crescent-shaped area at the base of the human fingernail	a time when the Moon presents a particular recurring appearance
the time at which the Moon is at first or last quarter when half its face is illuminated