The lunar "day" is actually nearly a solar month and it makes sense to call that a month. However on a shorter timescale, it makes sense for the "day" to be the

The lunar "day" is actually nearly a solar month and it makes sense to call that a month. However on a shorter timescale, it makes sense for the "day" to be the interval for one rotation of the earth relative to the moon. So at noon each day, the same point on the earth is facing the moon, say the Greenwich meridian. So this lunar day would be slightly longer than the earth solar day, roughly 30.53/29.53 earth days or 24 hrs 49 minutes approx. Would make sense to call such lunar days something different like "earths." So you would say something like "5 earths until the sun sets." Makes sense to also keep day hour minute and second the same as earth synced with UTC.