To reply to the question of what these songs are called, the only term I have come across is RANT. MacPherson's Rant is the commoner title in Scotland, and I have come across one or two others I can't name right now. Strictly speaking, a rant should be an outpouring from the heart against some supposed injustice (like being hanged unnecessarily or unjustly), so if repentance is expressed in the song it would have to be called something else. Unless, of course, such repentance is to say "I'm sorry, but hey, it shouldn't end like THIS"