The second youtube link you included is taken from the 1976 album, Songs of Rebels and Redcoats (National Geographic 07788, 1976). Here's some info from the liner notes. Sadly, still no info about a particular version's origins.

THE GREEN MOUNTAINEER
On May 10, 1775, three weeks after Lexington and Concord, a detachment of several hundred volunteers, including "Green Mountain Boys" from the area of the New Hampshire Grants, known now as Vermont, opened the way for a rebel invasion of Canada by taking Fort Ticonderoga, guardian of Lake Champlain, without firing a shot. Although Col. Benedict Arnold showed up with a commission from Massachusetts, main credit for the victory is given to the Vermonters and brawny Ethan Allen, their salty commander, who planned and executed the surprise attack. As part of Gen. P. J. Schuyler's army, the Green Mountaineers shared the defeat at Quebec, but were later victorious at Bennington under Allen's cousin, Seth Warner.

The singing is the same as the version by Dorothy Mesney quoted above, but with the addition of the following stanzas adapted from Whittier's poem, as you pointed out:

On the South came the Hessians, our land to police
And armed for the battle while canting of peace
On our East came the British, the red-coated band
To hang up our leaders and eat up our land

Ho - all to the rescue! For Satan shall work
No gain for his legions of Hampshire and York!
They claim our professions, the pitiful knaves
The tribute we pay, shall be prisons and graves!