Agreed, to an extent. I don't read Greek so I can't read the original, but the context clearly implies atheist or agnostic, and at least one translation uses atheist. So, at the very least, it squarely rebuts the premise of the NYT article.
IIRC, the relevant portions I had in mind are Book 10 of The Laws. It seems the translation I read that uses atheist literally I only have in hard copy in a box somewhere (I only have The Republic in both hard copy and Kindle format[1]), so can't quickly identify the specific paragraph and provide a reference to other online translations (and the original Greek). But here's an example excerpt from Book 10 at http://classics.mit.edu/Plato/laws.10.x.html that makes the meaning clear:
> For do but mark the effect of their words: when you and I argue for the existence of the Gods, and produce the sun, moon, stars, and earth, claiming for them a divine being, if we would listen to the aforesaid philosophers we should say that they are earth and stones only, which can have no care at all of human affairs, and that all religion is a cooking up of words and a make-believe.
Also, to reiterate smogcutter's point, in the absence of counterveiling evidence, we should assume that our meaning of atheism (the concept, not necessarily the literal term) and of similar beliefs along the spectrum of divinity and mysticism are substantially similar to their meaning at any previous point in time, modulo the finer details, implications, etc. There is scant evidence that basic concepts are gatewayed or shaped by language or cultural context; at best the evidence only shows that language and culture work at the margins, modulating significance, popularity, etc.[2] More generally, we have a tendency to dehumanize and belittle people that we cannot identify with or only understand in caricature, and people from antiquity and earlier are perfect examples of such people. When we have the wherewithal to recognize the situation, our default, baseline assumption should be that one person's experience is the same as any other's, at least in the broad strokes.
We should of course be open to contrary evidence, and restrained in how we extrapolate. But in this particular case of the existence and import of atheism, there isn't any, AFAIK. What evidence there is bolsters the argument that the concept not only existed but was substantially similar, regardless of popularity. Even if you don't take the literature at face value, it strains credulity to believe people could conceptualize abstractions so close but that nobody could arrive at the idea that a) gods don't exist without b) substituting gods with some other mystical component instead of being contented with the unknown. Undoubtedly many, perhaps most--including Socrates and Plato--did perform the substitution, but surely some did not. Today you don't get published for the absence of any answer; you get published for providing firm, provocative answers. I'd bet the same was similar back then--nobody copied a scroll that says, "The gods don't exist. Full stop." You get copied for saying, "The gods don't exist. Instead, the universe was created by [insert provocative theory]."
FWIW, I don't identify as either atheist or agnostic. I took my adolescent Catholic confirmation vows with eyes wide open, including the understanding that it would always be a struggle to maintain them and to maintain a distinction between my analytical, empirically-grounded beliefs (and non-beliefs); my culturally American beliefs about equality, fairness, and respect for diversity; and my mystical beliefs and practices. But that in no way means I can't or don't fully understand or appreciate modern atheism and its implications and benefits. It doesn't even mean I can't believe in it, in the sense of applying it practically. So, for example, that Socrates was a devout worshipper isn't particularly persuasive on the question of whether Socrates could or did conceptualize of an atheism as we understand it. What's persuasive is that in making arguments that walk like a duck and quack like a duck, he may have had in mind a duck. Applying something like smogcutter's principle: if I can hold onto and apply a complex, multifaceted, and seemingly contradictory set of beliefs and concepts, the most intellectually honest and morally humane presumption is that Socrates, Plato, and others could, as well.
[1] Searching for "atheist" in Harold Bloom's translation of The Republic returns a hit for footnote 6 of Book IV: "The charge of studying the heavens was a serious one; astronomers tended to be atheists and were accused of so being. They gave mechanical accounts of the movements of the heavens, denied that the Sun and the Moon are gods, etc. They sought out what should be left inviolate. In the accusation of Socrates heavy weight was given to the allegation that he studied the heavens; it is not casual that this speech is presented as an apology." Bloom is a controversial figure, and I'm not trying to use this footnote as evidence for usage in The Laws--that context speaks for itself, IMO. I'm just pasting this here because, at the very least, it shows that there are alternative interpretations of the extant material than those cited in the Wikipedia article you mentioned. And in any event, arguing that atheism was a broader and more imprecise label in antiquity is different and far less dubious than arguing that the stricter concept the label implies today was non-existent back then. The certitude in the cited scholarship strikes me as denigrating. It's one thing to emphasis and appreciate cultural diversity, especially as a way to avoid biased analysis, but when it implies an intellectual or cognitive constraint alarm bells should ring.
[2] See, e.g., the Sapir-Whorf hypothesis and associated research. AFAICT, the weight of evidence is substantially against the original, strong hypothesis. The supporting research is thin and weak, if not debunked entirely. It only seems to survive in a substantially weaker form--the dynamic exists, but not in a way that justifies the original thesis that language gateways critical concepts.