Eli Bashwinger has a point, and there does indeed seem to be something wrong with the quotation.
I think the article makes an implicit distinction along these lines: a presumably modal notion of consistency applies to a set of uninterpreted sentences, whereas a non-modal notion of consistency would apply to a set of interpreted sentences; where a sentence is interpreted when it is assigned a truth value, when all its variables, if any, are bound by quantifiers, when quantifiers are assigned domains of objects, when predicates are assigned extensions, when logical constants are assigned referents, etc.
Given a distinction between interpreted and uninterpreted sentences, we may distinguish modal and non-modal consistency as follows:
Modal consistency. A set of uninterpreted sentences "is consistent if and only if it is possible for those sentences to be jointly true."
Okay. But what seems to be missing from the article is the idea that the right-hand side of the biconditional may in turn be further analyzed like this: it is possible for a set of uninterpreted sentences to be jointly true if and only if there is an interpretation or model in which those sentences are jointly true. (A given possible world may provide such an interpretation.)
Non-Modal consistency. As Eli Bashwinger puts it, a set of (fully interpreted) sentences is consistent if and only if the sentences are jointly true (at a given world.)
Now that we have a tentative distinction between modal and non-modal consistency, we have to ask whether or not Kripke's and Lewis's accounts of possible worlds turn on this or any other notion of modal consistency, and whether this is a problem if either does.
I'm not sure about Lewis's account. Kripke's, however, does indeed seem to turn on some modal notion of consistency as applied to sets of uninterpreted sentences.
What is unclear is whether or not this is a problem for him. On the one hand, yes, modal consistency appeals to some notion of possibility, and so this may look like a problem. But, on the other hand, the notion of possibility to which it appeals is the notion of there being an interpretation in which a set of sentences come out jointly true. In other words, Kripke doesn't seem to appeal to a primitive notion of possibility, but to one which is already analyzed: existence of an interpretation in which sentences are jointly true. The structure of the argument is unclear to me, so I am not sure whether this is a problem for him.