Did the ancient Greeks understand that the moon shines by reflection of sunlight, and that this was the explanation of its phases? Did this allow them to conclude that the moon was a sphere? From what little I know of ancient Greek science, I thought they didn't actually understand the notion of seeing by reflection of light.
The explanation of Moon phases (and eclipses) is basically the same under all theories of vision that reduce to geometric optics, because under all such theories illumination of the Moon is due to sunlight. The round shadow on the Moon during eclipses was used as one of the arguments for spherical Earth (e.g. by Aristotle in De Caelo), but of course it is hardly conclusive. Spherical Earth was accepted mostly because it fit into the "perfection of cosmos" doctrine that originated with Pythagoreans and Plato, but the round shadow didn't hurt. Anaximander was the only one who went with a cylinder.
Details of vision theories vary, but perceiving incoming light, including reflected light, was one of them circulating in antiquity. It is now called the intromission theory and was favored by Aristotle and the atomists. It was not the most popular one however. Most pre-Socratics, Plato, and geometers and astronomers from Euclid to Ptolemy seemed to prefer the extramission theory, where the visual rays coming out of the eyes "feel" distant objects, or some mixture of the two. Plato for example writes in Timaeus:"...whenever there is daylight round about, the visual current issues forth, like to like, and coalesces with it and is formed into a single homogeneous body in a direct line with the eye". Surprised? Don't be, according to 2002 paper Fundamentally Misunderstanding Visual Perception "...when adults were repeatedly asked specifically to draw whether something comes into or goes out of the eyes when person sees a balloon, 69% placed outward-pointing arrows..."
But I digress. Even under the strict extramission theory what visual rays "feel" depends on what is there, and that depends on other sources of light, like the Sun or a lantern. If they 'light up' other objects that is "felt", and if the Sun is badly positioned, or the Earth is in the way, that is "felt" too. Empedocles, one of the first extramissionists, explicitly likens eyes to lanterns, Ptolemy is less poetic:"Illumination by itself, however, is a sort of excess-condition in luminous objects, so it hurts and offends the [visual] sense. Illumination is also created along with coloring in objects that are struck by light from outside." See Ptolemy and the Foundations of Ancient Mathematical Optics.
According to this, we don't know who it was to explain the phases using a spherical model, though it was before 600 B.C:
The first person to correctly explain the phases of the Moon is lost in history. By the time Pythagoras wrote in 600 B.C., the ancient Greeks knew that the Moon is spherical and that it revolves around the Earth. The Greeks understood how that motion causes the monthly changes in the Moon’s appearance. In fact, they even had pretty good measurements of the Moon’s relative size and distance from the Earth.
I dread going to Wikipedia, but it is helpful:
The ancient Greek philosopher Anaxagoras (d. 428 BC) reasoned that the Sun and Moon were both giant spherical rocks, and that the latter reflected the light of the former.
One of the sources cited there is from here, which has biographies of scientists; it has proved useful to HSM in the past.