"COnsider also the Cuddy Wren (or Cutty Wren). I have heard it asserted that it was published in written form in the C13th. That would (probably, depending on the view taken of its subsequent transmission) mean it was not a folk song."

There's a common misconception embodied in this quote, ie. that if a song has a known author it can't be considered as 'traditional' - not so (this is romantic twaddle)! In fact a surprising number of 'traditional' songs have known authors. For example quite a number of English trad. songs are known to have had their origins in 17th or 18th century plays. Additionally there's lots of evidence that many songs had a broadside origin. Some of the broadside publishers may have just been printing songs from an earlier oral tradition but some songs were probably newly composed when they were first printed (hence must have had authors - albeit anonymous).

The point about 'traditional' songs is that they have largely been through a PROCESS involving oral transmission, selection and change. And, of course, since Cecil Sharp first came up with this 'evolutionary theory' of folk song it has been endlessly debated, debunked, modified and qualified - I'll leave you to find all the references and make up your own mind (please, though, avoid romantic - largely made up - twaddle).