Ok. Fair enough and I've noted it. So, what difference between these two forms of egalitarianism do you think should be emphasized?
Well, let me caveat up front that I'm not a philosopher, and my primary source here for philosophical egalitarianism is the link you provided a few posts back, in which you particularly seemed to want to highlight a Marxist analysis.
But theological egalitarianism (in the sense relevant to this forum) has nothing to do with Marxism; it's not primarily concerned with economic questions (although it has implications for women's ability to participate fully in whatever economic system prevails); and while it is putting forward a moral or basic equality between men and women, it's doing so on theological and Scriptural grounds.
Your link stated that "contemporary egalitarianism is identified through its
substantive commitment to structuring social institutions in an egalitarian fashion—avoiding inegalitarian outcomes, and promoting equality of wealth, welfare, or power."
I'd say that theological egalitarianism is looking at inequality between the sexes, particularly in the household and the church (the two spheres where those inegalitarian outcomes are most fiercely promoted) but also in wider spheres such as education and employment, and saying that this inequality is incompatible with the gospel; with the good news of the kingdom of God. That it is not God's will that women be denied their full dignity and agency as human beings, full participation and ability to express their gifts and contribute to their community, full permission to pursue their God-given vocations, due to an accident of biology.
That's not quite the same thing as avoiding inegalitarian outcomes. We're not saying that everyone comes out of a cookie-cutter, that everyone should all do the same things, participate in the same way, take up the same roles, experience the same outcomes, without distinction; but that there are different gifts and vocations and ways to participate, and indeed outcomes, and that none should be closed to women
because we are women.
To go back to the OP, and the linked article, and the claim that men are attracted to weakness; that rests, bluntly, on an ideological normalisation of gender hierarchy. The idea that it's "natural" for men to be in charge, and women to submit, and that both men and women will be most fulfilled if they align their actual lives with that natural order of things. You don't have to look far, even on CF, to see many who have bought into this ideology and theologised it.
In that sense, I think we're at least one step back from philosophical egalitarians; your link describes this work of "advancing arguments against the natural inequality thesis" as if it is accomplished, and in the past; something to which contemporary egalitarians are indebted but not having to work at now. But that's not where theological egalitarianism is at; this particular sub-forum was created to give us one place on CF - just one! - where people couldn't, for example, come in and attack women in church leadership as rebellious Jezebels twisting Scripture to usurp men's rightful place. Because it's constant and exhausting everywhere else.
I don't know whether that's at least a helpful place to start; but it might at least give a little more context for the flavour of replies the thread has been getting.