While I agree with your points, the truth is, despite the discrepancies, they are constituent members of the Lunar Duo.
While most of the world didn't, at least till Gen 5, count "mythical" and "legendary" Pokemon as separate, it made sense to some degree. Only recently, the way the Pokemon are marketed is cut and dry between "normal" Pokemon and Event Exclusive "Mythical" Pokemon. That doesn't necessarily answer anything, but it is worth noting that the developers felt the more offensive based Darkrai would be too strong to be obtainable in-game. The same case is seen to some extent (despite obvious power differences) with Arceus. The truth is, despite being nearly as strong, Cresselia is still weaker, both in typing as well as one-vs-one against Darkrai, and doesn't share the same kind of usage.
Also, from a marketing sense, this makes sense. Darkrai is the "cooler" of the two, so people would be more inclined to go to a special promo event for Darkrai as opposed to Cresselia, which benefits the corporate side of it more.
There is another thing I'd like to add:
"Legendary" and "Mythical" Pokemon exist only outside of the Pokemon World / Lore. Within the "actual" Pokemon World, these terms have no meaning; Darkrai and Cresselia, like most other Legendary/Mythical Pokemon, are only there to guard the realm or maintain peace and balance.
Hence, in the development of the lore of the Lunar Duo, it makes no difference whether one is legendary and the other is mythical or not. This only makes sense from a marketing point of view, and so that is probably the reason for the distinction.
Furthermore, the points are anomalies to some extent, but don't prove that they are not part of the Duo. Even Pokemon folklore and game plot lines mention the two as well connected/ closely related to the activities of each other, even going so far as to directly fight against one another. Details like "gender" aren't all that important to the mechanics, as neither can breed anyway, and as for typing, the Psychic type has always been connected with the Moon (Jirachi and Lunatone are Pokemon that come to mind, as does Musharna).
If you have noticed, despite the stronger "connect" of the Fairy type to the Moon, there are few instances where Psychic has been converted to Fairy, specially true in cases of Legendary Pokemon, because it can cause inconsistencies in the backstory (or lore).
So, despite the term itself being Fan made, the connections are hard to miss, and they are still a legendary duo.