The orbital plane of the moon about the earth is inclined by about 5 degrees from the plane of the ecliptic, the plane that contains the mean earth orbit about the sun. So, some times the moon is above the plane of the ecliptic and sometimes below. When it is above, the moon appears to be lit from below and when it is below the ecliptic it appears to be lit from below.

The earth and moon move in their respective orbits with the sun at the centre of the solar system. If we imagine this in a three dimensional way we will see that at some time during its orbital movement the moon may be in a higher plane than the sun with respect to the earth. In this case the bottom part of the moon will receive light from the sun. That part will then be visible as a half-moon.

For the purposes of this question the Sun, the Earth and the Moon can be taken as moving in the same plane. The apparent motion of the moon around the Earth is slower than that of the sun, which is why we usually see only part of the Moon sunlit. Whenever we see an incomplete Moon, then either the visible or the invisible part is either in the East or the West half of the Moon, and only in the East or West half.

Both Sun and Moon travel from East to West and rise in the east and set in the West. This means that when the moon sets in the West, if its West half is the main lit part, that half will reach the horizon first and be at the bottom as it sets, as seen by anyone on the Earth's surface facing West. When it rises next morning, then unless the moon now is full, the lit half will rise first and anyone facing East will see the lit half as being on top.

Such a Moon incidentally, will be waxing.

A waning moon will have its Eastern half lit. When it sets in the West, the unlit Western half will reach the horizon first, with the lit Eastern half above it, as seen by someone looking to the West. Correspondingly, at dawn, the lit Eastern  half will be the last part to appear above the horizon and will have its round end down.