I have no technical knowledge about clocks at all, but this sentence sounds odd to me, shouldn't there be a lot of different clock technologies that work in different ways?
Or are they saying this because there is only one "best" technology for NASA purposes and that one is off by 56 us on the moon?
According to special relativity clocks on the Moon will run slower because the moon is orbiting the Earth, so the clock has a higher relative speed. However, general relativity says the clock will run faster, because the Moon has a smaller gravitational field than Earth.
In this case the second effect will dominate, so even a perfect clock will run faster: it's not the clock which is fast, it is time itself. GPS satellites have to deal with the same issue.
I wonder if, in the future, we will have a group of timekeeping satellites orbiting our sun. Maybe their orbit could be tweaked to have identical time dilation to Earth, and everyone can just sync off that.
Maybe we can derive some time keeping mechanism from the black hole at the center of our galaxy, or maybe we will just realize that our obsession with timekeeping is a silly terrestrial habit, and not really important at scale.
The bigger problem is that our timekeeping is earth based. The time definition should have been based on deep space, with corrections for time on earth.
So earth clock are slow with respect to clocks in deep space. Clocks on the moon are less slow are therefore faster than on earth