(I am working on a datapack for a server, with server-side only Fabric mods.)

Loading the server with the Fabric API included causes my datapack's custom dimensions to not load correctly; the worldgen and mob spawns are correct, but the biome listed in F3 and the visual effects do not work.

It used to work correctly when I was developing the server in the 1.17 snapshot 21w14a (with API version 0.32.7).

You don't say what version of fabric api you are using, there is a recent fix in this area:
#1571

This was released 3 days ago:
https://www.curseforge.com/minecraft/mc-mods/fabric-api/files/3437870
but the latest at time of writing is 0.39.2

Failing that, it is impossible to answer your question without more concrete information.
Just posting a screenshot and saying "look its different" doesn't help without psychic abilities. :-)

Apologies, my bad! Yes, I was already using 0.39.2, which has this issue. I have since done further tests. There were no errors in either the client or server log relating to this issue.

I spent a few hours trial and error-ing and it was caused by the fabricDynamicRegistry.dat file. Deleting it and restarting the server fixed it.

That file exists to try to work around this vanilla bug:
https://bugs.mojang.com/browse/MC-202036

That you are still this bug maybe a sign that it is not fully fixed?