A Peculiar Moon in Lord of the Rings

Lord of the Rings, the fantasy trilogy by J.R.R. Tolkein, takes place in Middle Earth, which has a different geography than our Earth, some different humanoids (Elves, Dwarfs, Hobbits) walking/talking trees (Ents) etc. But the weather, the vegetation and the animal life are mostly similar to ours. In particular, the sun, the moon and the stars seem pretty much the same. The characters are often out and about by night, so the phase and position of the moon is often mentioned. In one scene, however, the moon behaves a bit strangely.

In The Two Towers, Book Four, Chapter 4, Of Herbs and Stewed Rabbits, in the first page of that chapter, "The moon was now three nights from the full, but it did not climb over the mountains until nearly midnight, and the early night was very dark." The mountains are to the east of their location, which would block the early rising moon. Does "three nights from the full" mean the moon is waxing and it is three days before it will be full - or does it mean the moon is waning and it is three days after the full moon? I intuitively interpreted that as the first case: the moon is waxing, it's before the full moon. In Moon Phases in The Lord of the Rings there's an impressively detailed accounting which agrees with that interpretation. But when the moon is waxing, it will typically reach its highest point in the sky before midnight. This really only depends on the fact that the moon is illuminated by light reflected from the sun. In much of the rest of Tolkein, the moon does behave quite normally, for example: "If the Moon gave enough light, we would use it, but alas he sets early and is yet young and pale".

Here's another loony scene however. In the chapter In the house of tom bombadil: "In the dead night, Frodo lay in a dream without light. Then he saw the young moon rising;". Dreams are notoriously full of nonsense, so I guess we can't hold Tolkein responsible for unfortunate lunar behaviour in the bad dream of a little guy with hairy toes.

In societies without widespread artificial lighting, most people were very conscious of the phases of the moon. Today however, this is far from true. Mention the easily observed fact that the full moon typically rises around the same time as the sunset and then sets around sunrise and the expression on people's faces will all too often plainly reveal that this is news to them.