Catalin Environment fog doesn't know anything about the player. It just knows about the camera so it's always rendered in respect to camera.
What's in that screenshot is a classic, ages old, distance fog based on camera distance. In Godot, you enable this by adding a WorldEnvironment node to the scene, adding an Environment to it and enabling "Fog" there.
For this type of fog, its falloff can be computed as linear or as exponential. With linear, you get to control start/end distance range relative to camera, so you can have fog starting only at a certain distance from the camera, as seen in your screenshot. In terms of visual control, this is almost always preferable to the way the supposedly more realistic fogs like volumetric or exponential are handled. It's actually the most basic type of fog you can have, computationally and historically speaking, but still most useful imo.
Strangely enough though, Godot only implemented classic exponential fog, up until 4.2 or 4.3. I even complained about this on github. Basic linear fog is so fundamental and so easy to implement that it just boggles the mind that they left it out for so long. They were probably so dead set on fancy "superior" volumetric fog that they just "forgot".
TLDR: You need at least 4.2 (or 4.3), enable Fog in the environment, switch fog type to "Depth" (which is classic linear falloff fog) and adjust the begin-end range and density curves.
Alternatively, if you use shaders, you can write to the FOG built in from the fragment function. In that case you can do whatever depth calculation you like. The drawback is that you need to do it in every shader you want the fog applied to, but it's quite manageable via a shader include.
