Some dictionaries give two variants of pronunciation (with /ɑ/ vs. /ɔ/) for words like: fog, log, loss, etc. I think that the variant with /ɑ/ has nothing to do with the cot–caught merger because those dictionaries give only one variant for words like law (with /ɔ/).
Is there any perceived difference (dialect, style, etc.) between such variants (e.g. /fɑg/ vs. /fɔg/ for fog)?
We Americans speak different dialects, and we don't all pronounce things the same way. This means that this vowel can be pronounced in two ways, even among people who don't have the cot-caught merger.
However, it doesn't mean that these two pronunciations are equally common. Use the first one if you need to choose between them; it's usually the most common—although it probably won't be in all regions.
You may notice that /fɔg/ and /cɑg/ are the first pronunciations in some dictionaries for fog and cog. This was deliberate on the dictionary-makers part. There are a lot of Americans who don't rhyme fog and cog.
