In the beginning, as I understand it, Jazz attained it's name from the red light district of New Orleans. The ladies of the evening had a penchant for wearing jasmine and would ask prospective customers if they were looking for a little Jas. The music that was played in many of these bordellos was made by Black people, it was largely ragtime and blues, but always present was the syncopation that first identified this music. These fundemental african rhythms were brought to america when america installed it's system of enslaving Blacks. With these slaves came drums and rhythm, and while I hear euro-fans claiming their scale and system of harmony to be at the root of Jazz, I don't hear too many country musicians tracing the history of the banjo.
Jazz is the direction African music took in america.
Foster must have been considered a bit of a thief himself since almost half of his material either celebrated Blacks and Black culture, or he was using the eubonics of the time in his lyrics. De Camptown ladies sing dis song, do dah do dah. That real difference was in the syncopation of Black music as heard on european instruments. The spelling of Jazz as I understand it came from a need to eliminate a problem that occured with the advertising of Jass. By their second recording they had changed their name to Original Dixieland Jazz Band.