You are correct that if you are on thee moon and have a cs-133 atom with you is second will take that many transitions. And if you do the same thing on Earth, a second will take the same number of transitions.

But things get weird when you are on earth and observe a cs-133 atom that is on the moon. Because you are in different reference frames, you are traveling at different speeds and are in different gravity wells time is moving at different rates. This means that a cs atom locally will transition a different number of times in a second from your point of view on Earth vs one you are observing on the moon.

And it would all be reversed if you were on the Moon observing a clock back on the Earth.

They already have to account for this with GPS satellites. They all have atomic clocks on them but they don’t run at the same speed as clocks that are on the ground. The satellites are moving at a great speed and are further from the center of the earth than us, so the software that calculates the distance from your phone to the satellite have to use Einstein’s equations to account for the change in the rate of time.

Relativity is weird.