https://www.allaboutjazz.com/which-came-first-jazz-or-baseball

I have found other earlier cites going through interviews of Roy J. Carew, born in 1875, who moved to New Orleans in 1903.

I have been doing a bit of research as well (I am a documentary filmmaker and a jazz drummer) for my next film on women in jazz and have come across a few things to add to your research.

The first is giving you credit where due: shortly after you published your book you sent info to Lewis Porter aiding him in his research which set as April 12, 1912 as the first published appearance of the word “jazz”. Porter writes:

“Ben’s Jazz Curve,” Los Angeles Times, April 2, 1912.
“Jazz” seems to have originated among white Americans, and the earliest printed uses are in California baseball writing, where it means “lively, energetic.”  (The word still carries this meaning, as in “Let’s jazz this up!”) The earliest known usage occurs on April 2, 1912, in an article discovered by researcher George A. Thompson, and sent to me courtesy of Dr. Cohen.
The page is hard to read, so I have retyped the text, with clarifying comments [in brackets]:
BEN’S JAZZ CURVE. “I got a new curve this year,” softly murmured Henderson yesterday, “and I’m goin’ to pitch one or two of them tomorrow. I call it the Jazz ball because it wobbles and you simply can’t do anything with it.” [That is, it’s too lively for them to hit it.] As prize fighters who invent new punches are always the first to get their’s Ben will probably be lucky if some guy don’t hit that new Jazzer ball a mile today. It is to be hoped that some unintelligent compositor does not spell that the Jag ball. That’s what it must be at that if it wobbles. [That is, he jokes, don’t confuse this with a drunken “jag.”]

Please also notice that in this very first printed use of the word, it is spelled “jazz.”
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My second note is more for Peter Earhard (as I have not read Dr Cohen’s book to know whether he makes the same claim/inference): Sidney Bechet does not actually say he dislikes the word “jazz” in his book Treat It Gentle and I believe your attempt to identify Bechet as one of the musicians who dislike the word is incorrect. Sidney, in his book, follows the statement, jazz is “… a name white people have given to the music,” with, “…it does not explain the music.” There is no anger or dislike expressed over the word, but he proceeds to explain that he uses the word “ragtime” or “rag time” to show how there is “a spirit right in the word.”

Additionally, Bechet did not actually write those words in 1960. He spoke them into a tape recorder, before he died in 1959, that was then transcribed, and published posthumously in 1960.

I believe now, that spirit does exist in the word jazz, and it most certainly has for many decades at least if one were to only use Art Blakey as an example having named his own group Art Blakey and The Jazz Messengers.

I cover many elements of jazz history, live performance, and the economics of gigging as a jazz musician in my documentary films “JazzTown” (AppleTV, Amazon Prime…) and “Who Killed Jazz” (Vimeo).