Regarding what you wrote here:
“Perhaps the explanation is that one should distinguish between preparing within the same day for that same day (for example, from Sabbath night for its daytime) and preparing from one day to another, even if they have the same holiness. The former does not carry the status of ‘preparation,’ and is therefore permitted; the latter is indeed considered preparation, and is therefore forbidden.”
As I understand it, it is permitted to prepare from one Sabbath for the following Sabbath, and that undermines this suggestion.

Not necessarily. It may be that preparation from one Sabbath for another Sabbath does not detract from the Sabbath that is doing the preparing, because it is the same thing itself (and not merely the same holiness, like Sabbath and Yom Kippur).

I just now thought of another possibility. For some reason these matters are not brought in practical Jewish law in the Tur and Shulchan Arukh, and in general the laws of preparation from the Sabbath to a weekday are vague and not clearly ruled. Except for the preparation laws at the beginning of Beitzah (Rabbah’s preparation principle), which is something a bit different.
In Maimonides, Laws of Sabbath 23:7, he brings this law, but it is framed in the context of various acts of repair or finishing (because of the prohibition of repairing a utensil; see the laws before and after). And this is what he writes:

“If a vine branch is tied to a pitcher, one may draw water with it on the Sabbath; but if it is not tied, one may not draw water with it, as a decree lest one cut it and thereby fix it. It is forbidden to scour silver vessels with natron because that whitens them in the manner craftsmen do, so it turns out that one is repairing a vessel and completing its work on the Sabbath. But one may scour them with soapwort and with natron, and similarly all vessels may be scoured with anything. And it is forbidden to rinse bowls and pots and the like, because it is like repairing them, unless one rinses them in order to eat another meal on that same Sabbath. But drinking vessels, such as cups and jugs, may be rinsed at any time, because there is no fixed time for drinking. And one may not make the beds on the Sabbath in order to sleep on them after the Sabbath, but one may make them from Sabbath night for the Sabbath day.”
+/Gloss of the Raavad/ “And it is forbidden to rinse bowls and pots and the like because it is like repairing them.” A.B.: All this is forbidden only because it is exertion for a weekday.+

Regarding pots and bowls, Maimonides understands this as a matter of repair, and the Raavad indeed comments that it is because one is preparing for a weekday (although the Maggid Mishneh writes that Maimonides agrees with that as well).
If so, it may be that according to Maimonides this is not a law of preparation at all, but a law of repair. Folding from before the Sabbath for the Sabbath is not repair, but preparation for use that same day. But when one folds for after the Sabbath, it already counts as repair, because it is not being done for immediate use (like selecting for non-immediate use). This resembles the Maggid Mishneh’s words, who identifies the two explanations: if there is repair here, then it is preparation; and in my way of putting it, if there is preparation, then it is repair.
But according to this, it still requires investigation what the rule would be regarding folding for the next Sabbath. Perhaps it really would be forbidden. Where did you see that it is permitted? As far as I’ve seen just now, this whole law was not ruled in the Tur and Shulchan Arukh. On the contrary, the plain sense of Maimonides’ wording (and of the Talmud) is that it is permitted for this Sabbath and forbidden for after the Sabbath and onward.
See also Shulchan Arukh, Orach Chayim 323:6 and the commentaries there.

Now I saw that this permission is indeed disputed (from the expanded notes to Peninei Halakha):
In the Tosefta at the end of chapter 13 it says: “One may not rinse dishes from this Sabbath for another Sabbath.” And so too in Raavan, Shabbat sec. 359. And this is also implied somewhat by the wording of Maimonides 23:7. However, in Rabbenu Chananel’s commentary (Shabbat 113b) he wrote that it is permitted to do preparation for the needs of another Sabbath. And there are manuscripts that do not include this prohibition in the Tosefta (see Orchot Shabbat 22:158). In practice, Tehillah LeDavid 302:6, Da’at Torah 3, and other later authorities wrote that it is forbidden to prepare for the next Sabbath.