All these threads eventually end up arguing semantics.

There are songs that are traditional in our family, that the world has forgotten.

So don't get bound down with definitions. Decide which you prefer, definitions or songs. If you prefer definitions, go find yourself a semantics forum. If you prefer songs get out there and sing whatever the hell you like. Those that survive may eventually considered to be traditional by philologists, just enjoy what you are singing and share your songs as much as you can.