On the Google homepage there is a fun little moon game today - 11/21/2024.
It tells me there’s such a thing as a “half moon” 
Maybe I’m being overly pedantic, but it’s first quarter and last quarter when we see the moon half illuminated.
“Half moon” refers to either a first quarter moon or a last quarter moon. It’s useful when it’s not necessary to specify the exact phase. For example, there are several places around the world called “Half Moon Bay” because of the semicircular shape of a bay. “First Or Last Quarter Moon Bay” doesn’t quite cut it.