As is well known now, the cleric was inspired, in large part, by the character of Dr. Van Helsing, as ably portrayed by Peter Cushing, in numerous Hammer horror films in the 1950s, 60s, and 70s. Consequently, the class's signature ability is not spell casting but its ability to "turn away" undead monsters, such as zombies, ghouls, and, of course, vampires.
As I noted in my earlier post on the subject, anti-clerics lack the ability to turn undead entirely in OD&D. This is a stance that, so far as I can recall, no other version of D&D adopts, not even Holmes, which otherwise does include OD&D's anti-clerics (or "evil clerics," as the good doctor calls them). AD&D, intriguingly, does not allow evil clerics to turn or destroy the undead, but they can neutralize or even command them. Moldvay/Cook/Marsh gives all clerics, regardless of alignment, the ability to turn undead, a stance also adopted by Mentzer.
Of all these options, I find the one adopted by Moldvay/Cook/Marsh and Mentzer to be the least satisfactory to me, as it makes all clerics equally efficacious against the undead, regardless of their deity (if any) and alignment. The implication seems to be that the simple fact that a character is a cleric is sufficient to have power over the undead. The AD&D position, meanwhile, is more coherent, inasmuch as alignment is significant, but it's also more dualistic (or perhaps "multipolar"), since it gives all alignments some power over the undead. OD&D, as noted, implies a "monopolar" reality, where Law, which is often equated with goodness, is the only force to which before which undead monsters flee.
Of course, in my Dwimmermount campaign, Law does not equal Good, or at least not all Lawful beings can be reckoned good ones, except in the narrow sense that they oppose Chaos, which seeks the destruction of all. The religion of Typhon, to cite a prominent example from the campaign, is a staunch defender of human civilization, a paragon of Law, but it also turns a blind eye to oppression and cruelty and supports strength as an indicator of rectitude. As in the LBBs, there are no non-Lawful clerics, excepting demon worshipers, which follow slightly different rules.
Given that all Lawful clerics, regardless of the deity they serve, can affect the undead, it seems clear to me that it is a cleric's Lawfulness, not his deity, that grants him this power, a position that carries with it cosmological implications I've embraced rather than shied away from in my campaign setting. Though there is no "church of Law," as we see in some early D&D materials, there is a Lawful cult, one that eschews the usual gods and personifies Law itself as a kind of Supreme Deity. This cult doesn't have clerics but rather paladins, who, interestingly enough, lack the ability to turn the undead (as in Supplement I), an oddity that I think helps to paint a delightfully muddled cosmology -- just the way I like it.