Yes, the moon takes about 27 DAYS to orbit. 27 DAYS = 27 spins of the earth. 1 spin = 1 Day/night
The movement of the moon itself affects moonrise time, position and the relative angle from the sun (that’s why the full moon appears once a month).
If you want to see these things, you should record the moon’s rising position every day for a month. The moon’s rising position will move slightly every DAY. It’s hard to recognize to normal people.
On the other hand, the earth spins once a day. We are on the earth. The sun, moon, stars move at the (almost) same speed **when we see them from the earth. **Think about a person on a turning swivel chair in a room. The objects in the room and everything will move at the (almost) same speed when he sees them from the chair. That’s why the sun and moon are synced when we see them.
If you want to make the moon’s 27 days movement, you should separate it from day/night movement, and it’ll be complicated. I don’t know the moon’s 27 days movement is really necessary for games.
Sorry for this lengthy explanation. No offense really.
Attached the image. Server looks good. I guess some brightness values are not replicated, or not assigned to the lights.