Almost every setting should work like this. The planes are generally only known to a few sages and powerful wizards. And most who know of them only as it relates to their plane. Your villagers and soldiers and even sages would be clueless on this matter. In the real world how much to do we know of even the size of our plane (universe) is highly debated. Same with the age of our universe. And then are their others. So yes dark sun residents not having knowledge of the planes and only having limited experience with a few that have for some reason interacted with their own plane and Jackie different names for them. Yep makes sense. And a plane is so big and vast they may be interacting with a part of a plane that greyhawk interacts with and it is so far away from where Greyhawk interacted with it that the races and civilizations involved may not even be aware of each other’s existence. Planes aren’t villages in a city. They have distance that dwarf crossing thousands or millions of galaxies and then some.I am only realizing now that multiverse planes are irrelevant to Dark Sun. The original Dark Sun setting doesnt even mention them. The concepts of them only appear indirectly and incidentally as parts of spells and in connection to the elemental spirituality of the Cleric.
At that time during D&D 2e, the desire to discuss the planes only mattered to the Greyhawk setting. DMs who wanted to transport Greyhawk characters to or from the Dark Sun setting, speculated about how to make room for the Dark Sun setting inside the Greyhawk setting.
The original intent was to play the Dark Sun setting, and the Greyhawk setting didnt exist. How to link the two was an afterthought.
A later Dark Sun expansion book, Defilers and Preservers, mentions a few details for how to handle spells and the like.
• The Black = 2e Plane of Shadow (which was once a source for illusion magic, but in 5e no longer exists).
• The Gray = replaces both 2e ether and 2e aster, and serves as a "limbo" for the dead.
Relating to the Greyhawk multiverse, the Gray is a kind buffer against both the ether and the aster. The Gray absolutely blocks access to the Aster, but there can be some passages ways between the Gray and the Ether − if a DM wants to hook Greyhawk and Dark Sun together.
The thing is, in 4e and 5e, both the Shadow Plane and the realm of the dead have merged together to become the same thing, namely Shadowfell.
In other words, to say the only planes that exist in Dark Sun are the Material Plane and the Shadowfell, and nothing else, is an accurate way to represent the original 2e Dark Sun canon within modern 5e D&D.
There is no important reason to distinguish the Black and the Gray within 5e. Both are Shadowfell. One can easily say any distinction is regional. The "Gray" is the parts of Shadowfell that the memory of the dead inhabit, and the "Black" are the parts of Shadowfell that are unpopulated wilderness. Done. Somewhere in the Shadowfell wilderness is the entrance to a demiplane called the "Hollow". Also done.
Material and shadow − these two are the only planes.
Because access to the Ethereal Plane isnt really a thing, the earlier mention of Clerics accessing the Elemental Planes is actually problematic. It is better to understand that these "Elemental Planes" are regions within the Material Plane, such as volcanoes and streams and upper atmosphere.
There are spells that mention "ethereal". As far as I can tell, the source material isnt really clear about how these spells work, because the Gray realm of the dead generally blocks access to the Ether. It seems to me, when a mage "goes ethereal" in Dark Sun, what is actually happening is, they are entering and traveling thru the realm of dead. In other words, the 2e "ethereal" is also the same thing as the 5e Shadowfell. Per 5e, the "ethereal" mage is actually "shadow walking". The Shadowfell echoes the features of the Material Plane, so the "ethereal" mage can still navigate the Material Plane normally while within the Shadowfell, even if the Shadowfell version seems more gloomy and neglected than the living plane.
Now because 5e has merged the 2e planar concepts into the Shadowfell, there is an unintended consequence for the Dark Sun setting. Because the setting is more conspicuously involving the realm of the dead, the spells can obsorb a creepy necromantic flavor that might not originally be there. Then again, the Gray is the realm of the dead, so magic that involves it or passes thru it is accurately necromantic.
This goes back to the original point, in Dark Sun the planes dont really matter. If a spell technically involves the Shadowfell, this normally has little consequence for the populations who inhabit the realm of the living.