> They mean local lunar time, for coordination between actors on the moon. Lunar day length and even, due to gravity, second length differ from Earth (i.e., time moves at a different speed).

These two sentences aren't related to each other. Time moves at a different speed on the moon than it does on Earth. But that only affects interactions between the Earth and the moon. Two actors that are both on the moon are both on the moon; their Earth-based clocks will work without adjustment if their goal is to coordinate with each other.