To me, this article conflates lunar elevation and phase. Together with the use of ambiguous terms "up", "down", "above", and "below" makes it difficult to understand the author's intent and what their software is meant to visualise.

Lunar elevation (aka "altitude" in celestial coordinates) is sidereal, so the maximum range is determined by the viewer's latitude. The moon's orbital plane is not perfectly aligned with that of the sun-earth so there's another component there, but this only varies +/- 5 degrees of the sun's declination.

Lunar phase is synodic, so there's no significant variation in the observed illuminated portion with the moon's elevation over one night.

The full moon occurs when the moon is on the opposite side of the earth from the sun, so it rises at sunset, transits the meridian at midnight and sets at sunrise, (+/- a few minutes for seasonal variation).

Here's a nice visualisation of the lunar analemna - the figure described by the apparent position of the moon over a month - from Mt. Laguna at 32.8 degrees north. https://www.hpwren.ucsd.edu/news/20250212/

I also found this page with a decent explanation of the apparent effects of orbital cycles. https://www.cyclecalcs.com/learn/synodic-sidereal.html#faq

*analemma