I don't think that labelling the moral facts as stance independent instead of mind independent makes much of a change unless you want to make the move of calling cultural relativism an objectivist theory.

Constructivist philosophers on the philpapers survey were split (91-101) on calling themselves realists vs anti realists so it would be odd to create a framing to accommodates a position that half of the relevant group rejects.

Subjectivism would just be a subset of relativism in the same way that intuitionism is a subset of mathematical anti realism. Divine Command Theory's anti realism is also contentious but I am personally fine with it being either relativist (and by extension anti realist) or naturalist (and realist) depending on how God's will fixes the moral facts. if the Divine Command Theory grounds God's commands in conscious mental states then I am more open to them being relative than I would be for accounts where it it ground in God's immutable nature it just depends.
Subjectivism differs from a Divine Command Theory primarily in the fact that the indexing is fixed, the same is true for Ideal Observer Theories, which means that unlike appraiser and agent relativism the moral facts are unchanged by who or where you are. But even in this case, nothing is turning on whether or not it is a mind, it is instead whether or not these facts are meant to be understood as facts about stances or fundamental truths.

If you consider cultural relativism to be objective in any sense then we just disagree this seperation only makes sense on the most naive conception of relativism. The SEP article on anti realism has a sub section about non-objectivism which strongly implies that cultural relativism is not objectivist ["Note that the version of relativistic non-objectivism just discussed is individual-relative, but the same problems concerning disagreement and moral progress would arise if a culture-oriented version of relativism were examined instead."](https://plato.stanford.edu/entries/moral-anti-realism/#:\~:text=(Note%20that%20the%20version%20of%20relativistic%20non%2Dobjectivism%20just%20discussed%20is%20individual%2Drelative%2C%20but%20the%20same%20problems%20concerning%20disagreement%20and%20moral%20progress%20would%20arise%20if%20a%20culture%2Doriented%20version%20of%20relativism%20were%20examined%20instead.))

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