Essentially, we have a dichotomy between two uses of the word 'traditional.' In one sense, we tend to use the term to attribute authorship when the original author is not known and, in another, anything which has has been passed down from generation to generation, especially orally. I don't think the second usage has anything to do with whether authorship is known or unknown.

I'd suspect that, especially in the venue of a traditional song contest that old songs that have stood the test of time and so became part of the 'tradition,' would be considered to be traditional. And, especially in the case of Burns, the popularity of his songs probably has more to do with their oral transmission from generation to generation than in the fact that they have been written down.