I think that the song's widespread appeal ~ as part of the "essential" repertoire of several different traditions ~ argues for its age and its traditional/anonymous origins.

Of course, it may have become so widely known and variously reinterpreted as a direct consquence of having been so successfully published and popularized ~ so the Countess certainly deserves a full measure of credit, whether or not she truly originated ("wrote") it, or simply discovered and disseminated it.