The really weird part to get your head around here is that in case of relativistic differences, it's actually time that's changing. So cesium still has the same frequency, the same number of events per second, but their second is not the same as ours.

When people talk about clocks running faster or slower, it muddies the topic a little because it makes it sound like a fault of the clock. The clock is accurately measuring the time it experiences.

My favourite trivia on this topic is that this is not hypothetical. If the clocks on GPS satellites kept earth time (eg, without accounting for relativistic drift), the entire GPS system would accumulate 10km of error per day. This is a genuine thing with measurable, real-world effects.

Nope, those are relativistic effects. This dude called Einstein had a few things to say about it: https://en.wikipedia.org/wiki/Theory_of_relativity

One of the effects in time dilation, where moving clocks are measured to tick more slowly than an observer's "stationary" clock.

I think the point is that the clock on the moon runs faster from the point of view of an observer on Earth due to time dilation. I can't claim to understand it, but clocks "run slower" the deeper they are in a gravity well. To an observer colocated with the clock, it ticks accurately.

This isn't a new problem, though. GPS satellites have atomic clocks, and they drift 7uS or so per day due to time dilation.

It does. Not for the observer on the moon, but on the earth.

Hafele-Keating experiment:

https://en.m.wikipedia.org/wiki/Hafele–Keating_experiment

Lunar gravity is about 1/6th of Earth gravity. Clocks slow down due to gravity (Einstein's relativity). Thus, clocks tick faster if there's less gravity.

Aside: we can measure this effect on Earth, between the foot and top of a mountain. Tom Scott has a video about exactly that.

Yes — relativistic effects are so fundamental that they affect every system/process (including atomic clocks ) and therefore any measurement scheme. That’s why we talk about “time slowing down”. The movie Interstellar has one popular depiction of this.

Because spacetime. Clocks run at different speeds depending on the gravitational field in which they operate. This is true for any two clocks, not just between earth and the moon. Satellites have to constantly correct their clocks for the same reason.

From earths perspective, yes.

It’s just that time really passes quicker on Moon (not just an illusion).

Relativity, different gravity well. Cesium is the same, actual time is different.