You are all there... or there abouts...
These are all versions of the same song...
But not the version that I remember (or at least remember bits of).
For e.g. The opening verse was

It was early one morning at the breaking of day;
The young cocks they were crowing and the farmer did say:
Arise, my good fellows, and work with goodwill,
For your horses need their bellies to fill'

So at four in the morning, we rise from our beds
And its down to the pump house to douse all our heads

...That is all I can recall of the begining.
What puzzles me is that I was never lucky enough to see the Watersons live, so I had to have heard it on a recording.
I have searched through all the Watersons CD's that I can find, but there is no trace of the song on any of them.
Someone out there must have a copy of, or remember, an LP - maybe a folk sampler - with the Watersons singing this song. Dunc