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If most Cimmerians remain in Cimmeria, they're most likely going to be affected by that environment, and given that it's described as a foggy, gloomy place, they're going to be lighter-skinned than Conan is described to be due to less sun exposure.
Of course the Picts live in a densely-forested region and are dark-skinned, so environmental factors don't have universal outcomes, but it's a matter of comparison between Conan and other Cimmerians.
The story you're referring to is "Shadows In Zamboula" . I need to re-read the story again, but I think you're correct. There are also mentions in the story "Queen of the Black Coast" , and others as well. For example, there is:
"And survivors of butchered Stygian ships named Bêlit with curses, and a white warrior with fierce blue eyes; so the Stygian princes remembered this man long and long, and their memory was a bitter tree which bore crimson fruit in the years to come."
and:
"The warriors fell into single file, creeping through the primordial twilights like black phantoms following a white ghost."
among others. It's possible that Howard referred to Conan as white in those stories in the sense of the 'white race', as in both he is among many black characters, but it's also likely his actual skin was being described.
There's a lot of wiggle room in Howard's descriptions, for example Conan's skin was often referred to as bronzed or browned from the sun, but bronze skin isn't that dark, and brown can cover a very wide swath of tones. Even then, given the qualifier of sun exposure, Conan would probably be much darker than most Cimmerians in Cimmeria.
I know the Zingarans, weren't portrayed as particularly dark, and they're supposed to be darker than Cimmerians. They're also supposed to be much lighter than the Picts, who would probably have skin tones comparable to Amerindians.
I haven't looked into it too deeply, but I think that Howard may have described Cimmerians in that manner as a mythological origin of the "black Irish", with which he felt some affinity.
IIRC The Black Stranger (title unrelated, the stranger was a shadow demon) mentions that a very beautiful woman was fair skinned, and 'could have been mistaken for one of the white races at a glance'. She was Zingaran or Zamorian I think. Shadows in Zamboula is also replete with examples.