Great work finding the 1808 "Unfortunate Rake."

The important thing is that it was apparently an established song air rather than a dance tune. So the song must have had to do with an unfortunate rake.

Of course, there may have been an unrelated (or vaguely ancestral) song by that name, but is there any evidence of it?

If not, the most likely explanation is that the song we're discussing was the one attached to it. It would be perfectly natural for the word "lad" to be replaced at some point by the more specific "rake."

Factoid: The American "Streets of Laredo" tune is clearly that of the Irish "Bard of Armagh" - another harper.