Two different records, either or both could apply. If both apply, perhaps that age 13 marriage was annulled, and she had a successful redemption becoming a teacher and traveling abroad:
Admitted to Cortland (NY) County Poor House 2 Mar 1875, aged 13. Her parents were "intemperate" (i.e. drunkards.) The remarks section of the admission form said: "This person is of miserable stock and was married by an injudicious Justice of the Peace. She is headstrong and needs restraint. May in time go out and support herself."
1908 passport application has Harriet F Potter declaring that she has never been married. She is employed as a teacher in Utica, NY
Note: No 1870s Census records have been found for this family.