Disclaimer- I am no music historian, just interested in music of all kinds (except rap and such-like) and read a bit about it as well.

Will Fly- Indirectly, you hit on one of the short-comings of Oliver's books. He does not trace the evolution of his subject. Morton, your example, in his early days is covered, but not his later development. In many interviews, Morton gave questioners what they wanted- tidbits about the bordello days, not all accurate.
Of course ragtime was important in turn-of-the-century N. O., as it was all over the country, and was part of Morton's background. Along with other N. O. blues-based musicians, probably all of whom played ragtime, he moved on to Chicago; others to New York, and abroad, where they developed and drew on many sources.

(One digressional point about ragtime- it was a very disciplined form, developed mostly by trained composers such as Joplin (classically trained), and later developed some quite free variations.)

Like many other N. O. musicians, Morton moved on, in his case to Chicago where musicians, black and white, gathered from all over the country. It was in Chicago after WW1 that jazz became full-fledged during the 1920s. Oliver, Armstrong, Johnson, Hines (classically trained), Billy Strayhorn (classical training), Dodds, Armstrong, Beiderbecke (classically trained) to name a few. Others went to NY, but Chicago was an informal college where musicians of varied backgrounds and interests came together. The term 'jazz' (many unsupported speculations as to origin of the word) came into common use there.

Jazz has continued to diversify, as others have pointed out, drawing in elements from all over the world.