:No, I don't think this is correct.  The point of an interpretation is to assign meaning.  In a Tarskian structure, the meaning of a predicate symbol is its extension, but Tarskian structures are not the only possible interpretations.  Even within very conventional 20th-century math logic terms, for example, the predicate "is a set" has no extension, because the collection of all sets cannot be a completed totality.  Nevertheless set-theoretic utterances, in which the variables range over all sets, have meaning. 
 :The word interpretation is more general than the strict sense you're giving it, and the article needs to reflect that.