This is totally off the cuff, and possibly I haven’t thought about this carefully enough, and certainly haven’t crunched any numbers, but in order for someone on the Moon to see lights on Earth, or vice versa, they have to be looking at the dark side of the object. A quarter or crescent won’t do, I think, because the light from the sunlit side of the object would much, much brighter than the pitiful artificial lights. However, when we are looking at the dark side of the Moon (new moon) the Sun is always in the sky, and the same would be true of people on the Moon looking at the dark side of the Earth (new earth, which is when the moon is full in Earth’s sky.)
The best chance we’d have to see city lights on the Moon from Earth would be a total solar eclipse (and the corona of the Sun might be too bright) or a lunar eclipse (and the light that refracts through the Earth’s atmosphere and gives the eclipsed moon its reddish hue might drown out the lights, also.)
If you were on the Moon, you’d have a chance to observe the dark side of the Earth in all its light-polluted glory during that lunar eclipse, when the Earth passes in front of the Sun from your point of view. However, a red ring would appear around the limb of the planet (the aforementioned refracted light) and I don’t know how easy it would be to see any city lights in contrast to that. During a solar eclipse, you’d be SOL; the Moon’s shadow is very small on the face of the Earth, and the vast majority of the face of the planet would be illuminated by sunlight.