True

> The point is relatively close to the moon, and changes enough to not collect natural bodies.

This is awkwardly written. Let's get some background first: the Lagrange points are derived by assuming the Moon's orbit around the Earth is circular. In actual fact it's elliptical, so the distance from the Earth to the Moon changes and consequently the distance from the Moon to L2 changes. The distance from the Earth to the Moon averages 384,400km and at that point L2 is 64,000km away from the Moon.

But the reason no natural bodies (like asteroids) collect around L2 is because it's unstable, not because it moves.

> L4 and L5 are way out from the Earth Moon system

L4 and L5 form an equilateral triangle with the Earth and the Moon, meaning they are (on average) 384,400km away from both the Earth and the Moon at any given time.

> and are very stable.

No, they are "meta-stable." If you drift away from L4/5 you will not be drawn back to the Lagrange point itself, but you can orbit those points without extra fuel. We see asteroids gathering at Sun-Jupiter L4/5, so that suggests they are weakly stable, but when you do the linearized stability analysis of those points you find that they are neither stable nor unstable.

> The actual math is pretty complex

It is. I made an effort to create some educational videos on the topic: https://www.youtube.com/watch?v=8q9in-VfpUM

https://en.wikipedia.org/wiki/Kordylewski_cloud

I believe you’d have to avoid the point itself and orbit around it.

Edit: apparently, there are two Earth Trojans:

https://en.wikipedia.org/wiki/List_of_objects_at_Lagrange_po...

But JWST orbits the Sun Earth L2 point, so the moon is not directly involved. Of course it influences, but it’s different than the Earth Moon L2.