Howdy Pards and Pardettes!
Sorry, I do not know...
The Lake and Earp materials are complex, and with their partisan champions and detractors.
Meaning, was Earp an old man spinning Tall Tales, and Lake an easy Dupe? Or was Earp factually remembering everything correctly and Lake only wrote what he heard? Or was Earp a mix of fact and invention, and Lake "embellished" what he heard?
The S.D. notation may be "innocent" and just the way that Lake jotted it down in its "modern form" instead of say "Dakota Territory."
In reading letters and remembrances of "old folks' on their youth, it is not unusual for them to use "current" terms and words that were not in use when they were young "back then."
American "history" is filled with such things. One of the "biggies" is Lydia Boggs Shepherd Kruger's affadavit in the 1840's (long after everyone was dead) that local frontier hero Betty Zane did not make the "Gunpowder Run" during the Siege of Fort Henry in 1781 and that it was Molly Scott.
Turns out Lydia was correct. Scott made a run during the Siege of 1781 when Lydia was there. Zane made a run during the Siege of 1782 when Lydia was not at the fort.