Apparently it is a very common misconception that the phases of the Moon is caused by the Earth’s shadow. Not only among flat-Earthers, but the general public as well. The difference is that flat-Earthers will usually jump to conclusion, and have some outlandish interpretation about this.
In reality, the phases of the Moon happen because we see a different sunlit part of the Moon at different time. Not because it is over the shadow of the Earth.