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.
Like the Earth, the Moon is also experiencing day and night cycles. A half of the moon is sunlit, and the other half is dark. The lunar phase happens because we observe different sunlit part of the Moon depending on its position relative to the Earth. The lunar phase is constantly changing. It changes gradually over the period of about 29½ days.
On the other hand, lunar eclipse happens when the Earth’s shadow is cast on the surface of the Moon. This is a rare phenomenon. It can only happen for a few hours during a full Moon phase, but not every full Moon will have an eclipse. Every year, there are at least two lunar eclipses and as many as five.