Does anyone really refer to the crescent kick/half-moon kick as described in the 2011 KKW technical terminology by ahn and capener? They describe the bandal chagi as similar to dollyeo chagi, but supporting leg does not pivot, like an ap chagi; and kicking leg is slightly turned in to strike.
Most references i see to bandal chagi seem to take the older approach of it being an outward or inward kick (bakkat or an chagi), swinging the leg up to strike with the outer edge or inner arch of the foot.