As Logikal has noted, the post in question has a response that can be characterized as idiosyncratic. Anyone can philosophize in the broad sense, but professional philosophers and highly-educated amateurs gravitate towards conventions in language to express ideas. Imagine your surgeon showing up with her own names for bones and vasculature! Because it's not as obvious ideas kill people (though arguably they do), anyone can philosophize without a license (and should IMNSHO).
Roughly, ethics and epistemology are two very large branches of philosophy, the former being concerned with truth and falsity, the latter with virtues and vices. They are intertwined, naturally. Major theories of truth include those of correspondence, coherence, pragmatism, deflation, and disquotation. In ethics, there is meta-ethics, normative ethics, and applied ethics. Until you understand roughly what each is, it's easy to get confused conceptually. Roughly, epistemological concerns answer a question like 'what is true; real or not contradictory', and ethical ones revolve around 'what do I value and what should I do'. All about 'What is?' and 'What Ought?'.
The post in question attempts to create a conceptual taxonomy based on a simple permutation across two variables: degree of moral absolutism vs. relativism, and degree of subjectivity or objectivity. Now central to decoding this taxonomy are ontological questions. What is the nature of these moral ideas? Are ideas out "there" somewhere or they in "here". This is the issue of subjectivity and objectivity and is roughly correspondent to issues of duality, often expressed as Cartesian duality. Plato thought math ideas were out "there". George Lakoff believes ideas are in "here".
Note that though the taxonomy is non-conventional and muddled, some sense can be made of it, and your questions speak to questions of logical relations. In the peculiar nomenclature of the poster, as best as I can interpret:
- An absolutist subjectivist holds that there exists a universal human morality because human minds are essentially the same and construct it. This would be some form of psychological altruism, something akin to the ideas that E.O. Wilson proposes in his sociobiology (distinct from biological altruism) and that is embraced roughly by humanists.
- An absolutist objectivist holds that there are transcendent values that exist by some external cause such as a god or gods and exists independent of the human mind. Traditional Christian theology might be an example of this category. An all-powerful God, Jahweh, has determined right and wrong for us, and applies to all humans everywhere and at all times.
- A relativistic objectivist would hold that there are universal rights and wrongs, apparently either intrinsic or extrinsic to the mind, but contextually dependent as in situational ethics, but that the rights and wrongs are universal, say using the Kantian categorical imperative.
- An relativistic subjectivist would hold that there are rights and wrongs independent of people and that ethics is situational, but it depends on the culture as in cultural relativism and other flavors of post-modernist thinking. This last categorization seems contradictory because generally rights and wrongs presumed to be independent of the mind are from a single source, as far as I know, but one could see this as competing meta-ethical systems. I guess in this category, one could claim that both Greek and Norse gods are human-independent sources of right and wrong, but that Greek morality applies to Greeks and Norse mythology applies to the Norse, and that neither the Greek nor Norse gods have a privileged position. Plus, Greeks could be free to argue over interpretation? This seems sort of meta-meta to me.
So, the OP from the other site seems to have developed a personal taxonomy which is awkward because the two dimensions seem to my limited mind to conflate the various ideas of situational and social context, transcendentalism, and constructivism while blurring the lines between meta-ethical and ethical systems. The taxonomy is mathematically appealing, but conceptually confusing, which is likely why it's idiosyncratic and not conventional.
While the taxonomy is dubious, your questions are incisive.
Am I correct that in 1, 'relevant respects' refers to those facets or features relevant to 'moral facts'?
It depends on the ontological nature of how the relevant respects of the mind is defined. The can of worms here is the relationship between normativity (read value-laden subjectivity) and positivity (read value-free objectivity) of the mind. Is measuring purely objective? (It's not.) Does grammar and diction about reality influence the perception of reality? (To some extent it does.) Does math have a role to play in morality? (It does if you are a utilitarian.) This a can of worms because of debate over the fact-value distinction.
Doesn't 1 imply 2? Because 'everyone's mind is the same in the relevant respects' ⇒ 'everyone's mind differs in the IRrelevant respects' ⇒ 'we all have different minds'?
There is some stark black and white thinking in the dichotomy because the subjectivist position claims everyone's mind is the same, and the objectivist position presumes everyone's mind is different, and of course, both are true to an extent. This situation in philosophy is paradoxical and leads the discussion about dialetheistic logic which allows for contradictions to be evaluated at some sort of meta-level. The logical implication arises from a binary definition of classification that is all-inclusive and obeys the law of the excluded middle.
If the answer to 4 is yes, then is Absolutist Subjectivism a hyponym of Relativist Subjectivism?
Yes, conditional on your presuppositions about the validity of claiming there is a meta-meta-ethics. In this two-axis matrix (S,O)x(A,R), AS is not a type of RS because the criterion that differentiates membership Absolutism from Relativism creates a mutual exclusion by logical contradiction on a meta-ethical level. There cannot be ONE set of right and wrongs absolutely and then and have everyone have their own relative set. BUT, one could say that one's meta-ethical beliefs are themselves subject to a form of relativism. For instance, note a scientifically minded atheist could claim that the only valid meta-ethical reality is that there is no god, and all morality is a universal human morality that we more or less all share (with pathological exceptions), and yet another atheist could claim that it's meta-ethically permissible for less "developed" cultures to be allowed to continue in their "universal" meta-ethical belief that there is one true God and all else is blasphemy (Think prime directive from Star Trek). In this case, it would appear that the taxonomy is recursive such that absolutist objectivists on two distinct planets are both wrong (only absolutist subjectivism is scientifically correct), but that it's permissible for political reasons to maintain that the cultural relativism of subjective relativism is pragmatically true. (This is where theory of truth has bearing).
Remember, if one sees this as a two-axis continuum that utilizes partial membership, the discussion gets more interesting. As an economist, you should be aware of the philosophical difference between integers and reals and crisp and fuzzy categories.
So, like many philosophical questions that ask, is it logically implied, the question ultimately boils down to metaphysical presuppositions about what constitutes logic and truth, and what it means to be right and wrong. These are the issues that metaphysicians wrestle with.
That was an incisive question (meaningless and muddled questions are less fun to answer). You clearly have the hallmarks of a critical thinker. Keep up the good work!