Lunar Terminator Paradox

There's an optical illusion known by names like the "lunar terminator paradox". This has much discussion and descriptions online, but none made sense in my head, so I wrote this program to figure out what was going on. The paradox:

I was camping in the mountains with a friend recently. We were looking up at the sky, just after the sun has set. The moon was clearly pointing up.

After playing with this program, I have clarity: we're looking up at the moon from below. The most pathological case is a full moon. Let's say the moon is on the horizon: elevation=0. If we're looking at this low moon, and the moon is full, the sun is directly behind us. The moon is fully illuminated, and we see 100% of the moon disc lit up.

Now, what happens if the moon is not at the horizon, but higher up at the sky? Effectively, the sun is infinitely far away, so the same part of the moon is lit up. But we're now looking at the moon from below, and we would see some of the dark side on the bottom edge. I.e. the moon appears bright at the top, but has a dark slice missing at the bottom: it points up! Even at sunset.

This is the most clearly weird case. A half-moon has no paradoxical behavior, and the more full the moon, the more clearly we can observe the bright moon pointing up despite the sun being down.

I produce two plots. At 3/4-full moon at sunset we look at the moon as it rises from the horizon all the way up to the zenith: