I'm still wondering about the source of the Digital Tradition version. Maybe it's from the singing of Frank Warner himself. The Penguin Book of American Folk Songs (Alan Lomax, 1964) has a text that also comes from Warner. It's almost the same as the other two, but there are some differences. Note that this version does not have the final "Fight On!" verse, and neither does the version in the Digital Tradition.

The Dying British Sergeant

Come all you good people, where-e'er you be,
Who walk on the land or sail by the sea,
Come listen to the words of a dyin' man,
I think you will remember them.

'Twas in October, the eighteenth day,
Our ship set sail for Amerikay,
The drums and the trumpets loud did sound,
And then to Boston we were bound.

And when to Boston we did come
We thought by the aid of our British guns
To make them Yankees own our King,
And daily tribute to him bring.

But to our sad and sore surprise
We saw men like grasshoppers rise,
'Freedom or death' was all their cry,
Indeed, they were not feared to die.

When I received my deathly wound
I bid farewell to England's ground,
My wife and children shall mourn for me
Whilst I lie dead in Amerikee.