Still, different usages of the word 'communal' as you say. I could quite imagine the waulking songs being communal in that as they were doing that task it would easily lend itself to different persons involved in making up the verses. In that case I would say 'communal' would cover it. However, we are led to believe that the chantyman was the soloist and the others doing the work and singing the chorus, that's the nature of the beast.

The problem with this in our sphere here, there were endless arguments in the early 20th century amongst largely American scholars some of whom tried to claim that the ballads were made up communally, until the theory was eventually thrown out of the window. One of Child's students, Francis Gummere, was one of the first proponents and then the studies of Albert Lord and Parry in the Balkans lent weight to the theory. So 'communal' is a bit of a dirty word in folk research circles.

However I have come across songs that would qualify for 'communal composition', though a single person would still have come up with the tune and chorus. We have a very widespread and popular 'bothy' song in my neck of the woods that has so many wide-ranging verses involved that it must have been the composition over time of many people. I imagine at least some of the NE bothy ballads must have originated in this way. Also you could easily imagine that some of the more basic catalogue type rugby songs were created in this way.

Nowadays we only think of communal composition in terms of these songs having been passed through many voices and having been altered so many times that in that sense the latest version has been communally composed.