Joshua, unfortunately a multi-participant discussion risks simply confusing interaction, but I’ll contribute anyway. The idea of forming a “Genealogical Adam Studies” vocabulary is a good one, because done right it can save the kind of confusion that so often occurs. One linked example is how loosely the theological conceptions of “creation” map to the Hebrew use of “bara”, so that one’s never quite sure whether a biblical or philsophical view is being proposed.
The criteria for a vocabulary need, I think, (1) to be as closely linked to biblical categories as possible, so that our terms have some equivalence to Hebrew usage; (2) To be cognizant of the koind of scientific/theological issues it will need to cover, such as the spiritual status of ancient races and species; and (3) to be aware of existing biases in understanding - for example, “pre-adamite” has a baggage of implications going back to other debates and other centuries.
Therefore, some principles are that the English capitalisation of Adam for an individual or archetype is useful, but so is the Hebrew use of the article, making ha’adam what was created in Gen 1. So I’ll now list your list, with my suggestions following in bold:
1 “Adam” would be a real person a real past (Genesis 2), who we scientifically expect would become ancestor of all of us (if he lived anytime at or before 6 kya), but there will be debate about the theological importance of descent from him. I agree with this, because it lays out the basis of the genealogical Adam case - those who disagree will simply dispute the science. Diluting this leaves room for those like Patrick, who appear not to have grasped the issue, to debate what is irrelevant to the matter in hand
2 “adams” is the same biological type as Adam, and we might argue they arise as a community before Adam (Genesis 1) I would suggest a Hebraism, ha’adam, here, firstly because it is what Genesis and the rest of the Bible uses, secondly because the lower case “a” achieves the distinction you’re after, thirdly because it saves possible confusions like “all sinners are little adams”, and fourthly because it sounds swankily intellectual (did I say that??)
3 “Sons and/or Daughters of Adam” are the biological descendants of Adam (see Genesis 6) Yes - this could include phrases like “children of Adam” etc.
4. The “first Adam” and “second Adam” are referencing Adam and Jesus, the two adams (biological type) God (perhaps miraculously) created with a redemptive purpose (perhaps de novo = Virgin birth). Yes - existing biblical category into which all theological discussion can fit
5 “adams” are “Adamites,” in that they have the same biological type as Adam, even though they do not all descend from him (e.g. when they pre-exist Adam). This I find confusing, and I’d rather dispense with the word “Adamite” altogether, lest someone “usefully” coins “adamite” and merges all our categories into a mash. May I suggest, for this and the following point, that we coin a hellenism to distinguish mankind as described in Genesis from anything else. Hence I suggest the word eikonic to denote those who are created in the image of God, however we understand that. Ha’adam are then eikonic because Gen 1 describes them that way, and Adam is also eikonic (which nobody disputes). This enables a clear term to discuss the spiritual/creational status of Neanderthals and, indeed, all early hominids: they would be “non-eikonic hominids” (thus avoiding ambiguous terms like human/men etc). So, for example, people could debate whether the Neanderthals who hybridized with H sapiens were eikonic or pre-eikonic, whether “sons of God” in Gen 6 were non-eikonic hominids, non-Adamic or angels and, hopefully keep clear of different applications of the word “adam,” biological confusion about whether biological ancestors are adamites because precursors of Adam, or not
6 “Pre-Adamite”, would be all those before “adams” arise (e.g. “Lucy”). Though this is rather beside the point, because we are not talking about them in Genesis 1 or beyond. See above
7 “Non-Adamites,” as in non-Adamic beings alongside Adam who interbreed with his lineage, would not necessarily be an occupied category. Though, perhaps, in the @vjtorley scenario of Neanderthals are non-Adamites. This confusing already because of capitals and the “A” word. See above - call them “ha’adam” or “non-eikonic hominids/races,” and the theological issues are clear to work on. if one were discussing things biologically (say somebody discovered late interbreeding with Hobbits), the relevant species name could be used, with “ha’adam”, “eikonic”, or “non-eikonic” to discuss their spiritual status.
8 We can refer to the Sons and Daughters of Adam as a “new kind of adam.” Use our technical term, “a new kind of ha’adam” and the necessary pause for thought will save the tendency to confuse all the different nuances of “adam”, “Adam”, “Adamite”, “adamite” etc.
Hope that’s a helpful contribution. I’m just imagining scholars discussing this in a few years time when we’re gone: if our terminology is sound, so will be the development.