Weird Moon is when the moon is depicted in a weird way, either with weird eclipses, lunar phases, or size and shape. The definition of Artsy Moon states this can be "because of actual physical changes", while Weird Moon states that "There can be objects (such as stars) visible inside the crescent", and "it usually becomes an important plot point" qualifies for the trope. In other words, an actual physical change qualifies for both Artsy Moon and Weird Moon.
In addition, Weird Moon states that "The crescent moon is often depicted in a stylized, unrealistic manner", while Artsy Moon states the moon can be depicted this way due to Artistic License, which allows for a stylised and unrealistic depiction. All signs point to Artsy Moon being a subtrope of Weird Moon, with the only distinguishing factor being that the artsy moon is depicted that way for the "Artsy" part.
I've tried to write around this issue in the past by adding the text "The differentiating factor here is the "Artsy" part. If the Moon is depicted in a weird way for artistic value, then it belongs here. It it's just weird, or because it's clearly not our Moon, then it doesn't belong" to the Artsy Moon page, but what qualifies as "Artsy" is subjective. I've put examples where I think it is "Artsy" in Artsy Moon, and others in Weird Moon, but some examples I added, such as Shuna's Journey, is both Artsy and Weird. Yes. The moon is depicted in a stylised way, but the moon is also physically weird, so it blurs the line between these two tropes. The Tale of the Princess Kaguya, my other addition to the Artsy Moon page, does depict its moon in an artsy way (see page image for Artsy Moon), but there's also some elements of Weird Moon that impacts on the story, so it can belong in both tropes.
We either need to add a better distinguishing element (or explanation) to Artsy Moon to separate it from Weird Moon further, or merge the two tropes. See the page for Ridiculously Similar Trope.