I’ve always referred to it as a newspaper to distinguish it from their own sister publication “Harper’s New Monthly Magazine” which, being smaller, many more pages, and issued monthly, is a more definitive magazine. Early in its history the weekly called themselves a “family newspaper”, and modeled themselves against “Frank Leslie’s Illustrated Newspaper” which began about a year earlier. If Leslie’s was a “newspaper” then certainly Harper’s Weekly was a “newspaper” as well.
However, in Mott’s “History of American Magazines” he includes a section for Harper’s Weekly, as well as one for Leslie’s Illustrated Newspaper and other similar weeklies. Then he confuses the issue a bit more in the second volume of his book (pg. 43) by stating: “Half a dozen copiously illustrated weeklies of general appeal must be grouped separately. It would not be inappropriate to classify these periodicals as newspapers, since they all relied much upon the reporting of current events: indeed, one of them called itself a newspaper in its title. But they were all very much more than newspapers, and they placed the emphasis on features of appeal which belonged more characteristically to the magazine than to the newspaper–namely, pictures and belles-lettres…”.