Isn't the point here that in the video the shadowed part of the moon is on the top instead of the bottom as it would be to the observer in your diagram?
Ok, forget my first illustration.
Both the Moon and the Sun move in the same direction, so if the Moon is going down in the video, it means that it is becoming night.
The face of the Moon that is being illuminated is looking down towards the horizon, so the Sun at below the horizon and we can see its light being reflected by the Moon, so the shadowed part of the Moon is correct.
Also it is importand to know the location of the video, which the author says it is mexico. This is important because the Moon orbits roughtly around the Earth Ecuator, not from North to South, so someone in in the Equator will see the Moon in a different possition and rotation than other in Canada, respect the zenit.
For example, someone in Mexico will see the Moon pass almost above their heads, while I see in my city the Moon displaced to the west part of the firmament. Actually, where I live, i see the moon like this:
instead of
Hope this solves the problem.