daij, meikyo

I think this is a better x-ref.

@@ -12 +12 @@
-<xref type="see" seq="1184860">下弦の月・かげんのつき</xref>
+<xref type="see" seq="1353070">上弦</xref>

@@ -16 +16 @@
-<gloss>waning half moon</gloss>
+<gloss>waning half-moon</gloss>

https://www.timeanddate.com/astronomy/moon/third-quarter.html
The Third Quarter Moon is a Half Moon
By Anne Buckle and Vigdis Hocken
The Third Quarter Moon is when the opposite half of the Moon is illuminated compared to the First Quarter. Which half you see lit up depends on where you are on Earth.

https://starinastar.com/moon-phases-names-an-easy-way-to-remember/
Actually, I’d like to call the first quarter (waxing half moon) the “Earth’s tail moon” and the third quarter (waning half moon) the “Earth’s nose moon.” 

https://moon.nasa.gov/moon-in-motion/moon-phases
A last quarter moon, also known as a third quarter moon, rises around midnight and sets around noon.

Enormously common terms:
下弦	282732	30.8%
上弦	190064	20.7%
下弦の月	264063	28.7%
上弦の月	173615	18.9%
弦月	8330	0.9%

See many more comments/refs on 弦月, which may be updated after this.

Changed (moon's) last quarter to "last quarter moon", as this is a technical term.  It's not, generally, the whole "quarter" period of time. It is the period where 1/4 of the moon's surface is illuminated (1/2 of the surface visible from earth).  "waning half" is not super-technical, actually, but it's common enough, and the cleanest way to make it clear that this is a "half moon", which is both important to the imagery (bow-shaped) and the meaning of related terms (弦月)

daijs:
https://dictionary.goo.ne.jp/word/下弦/
満月のあとの半月。

On top of the mountain of stuff I posted, here is a news video explaining how to differentiate 下弦 and 上弦, and they are exact-half moons(not periods of time between phases).
https://www.youtube.com/watch?v=FtwKuKlJkd0

@@ -12 +12,4 @@
-<gloss>(moon's) last quarter</gloss>
+<xref type="see" seq="1184860">下弦の月・かげんのつき</xref>
+<gloss>last quarter moon</gloss>
+<gloss>third quarter moon</gloss>
+<gloss>waning half moon</gloss>