The confusion comes from this: John Uri Lloyd's famous novel, Stringtown on the Pike, was published in 1900. When Lloyd uses the name Stringtown, there's no doubt: he's referring to Florence. The City of Florence was never actually named Stringtown; it was strictly a reference to Lloyd's fictionalized version of the town.
A more interesting answer lies in the map, which refers to the area west of Constance (“Anderson's Ferry”) as “Stringtown.” The map is from a large book of Ohio River chart's that was undated. It did, however, say that it was “Reduced from a survey by. . .W. M. Roberts, US Civil Engineer, 1867-1868, with additions & corrections from later surveys, under the auspices of the Corps of Engineers.” So you can't tell exactly when it's from.
But be assured that the “real” Stringtown isn't Florence, but a small sliver of land just downriver from Constance, said to be so named by the river men who saw the little town in a string along the river.