I've been looking into whether or not Jack Girty was Snake's father. I have seen Ned and Daniel's applications unfornately the information really conflicts. The one you mention had the notation that Jack was the father added after the fact and I think it was just a guess on whoever wrote. Here's why.
From the research I have done, I find no reference to Snake Girty before 1861 when he served as a corporal in the Civil War in Waites regiment. He never appeared in the Old settler roll.
The 1896 payroll lists Snake Girty, but not as a descendant of Jack Girty. All of Jack's other children and grandchildren who were living were listed. Snake was listed as a nephew of a man named Old Ned and as an heir to anther deceased Cherokee, I don't have the name handy.
In either Daniel's or Ned's file it records the known relatives of Snake. It lists his mother and a brother (Indian name). It also has Snake's Indian name.
The 1902 roll lists the enrollee and who their parents were. For Snake it listed his mother and in the father block it said "unknown."
Then theirs Daniel and Ned's depositions. Ned Girty, a son of Snake Girty and Kah-u-gun-stah, related in his Dawes the following regarding his mother, "My mother's name was Ga-ha-gust; she was a sister of Wilson Girty; I do not know what relation she was to Buck Girty but think he was her brother; my mother's father was named John but I do not remember the rest of it; my mother's father always said that his father was a white man." Ned's brother, Daniel Girty, made a similar statement, "Ka-ha-gust was my mother; I do not know who her mother was but I think her name was Je-naw-sa." He goes on to say, "Wilson Girty was my mother's brother."
This would lead me to believe that Jack Girty (aka John Girty) was father of Kah-u-gun-stah and not Snake. So then theirs the question of why Snake's last name is Girty and what relation he really has with Jack. Well we can rule out that he married his sister. :)
My theory is that Snake Girty didn't have an English name before the Civil War and his father died when he was very young. He married Jack's daughter and, as I have read, he would have been required to take an English name in order to enlist in the military. Since he knew nothing of his father, who may not have had an English name, he took his father-in-laws last name. I have to get Snake's Indian name translated to English to see if that provides a clue. How this happened is of course my theory, but I think it was Snake's wife that was a daughter of Jack.
Of course more proof would be nice. I can send what records I have to substanciate this.