[Lordstan]

This is weird. The OP links to a SABR bio of Jackson, but even this reference refutes his claim about Jackson.

From the linked SABR reference

"The White Sox were rocked by the entry of the United States into World War I. Several Chicago players enlisted in the military, while others were drafted in the early months of 1918. Joe, as a married man, was granted a deferment by his hometown draft board in Greenville, South Carolina, but after Jackson played 17 games with the White Sox the board reversed its decision and ordered him to report for induction. Instead, Jackson found employment at a Delaware shipyard, where he helped build battleships and played ball in a hastily assembled factory circuit, the Bethlehem Steel League. Jackson was the first prominent player to avoid the draft by opting for war work, for which he was severely criticized in the sporting press, especially in Chicago. He won the factory league batting title with a .371 average, but the controversy permanently damaged his relationships with the Chicago sportswriters."

Notice, it doesn't say he worked for or was a member of the Army or Armed Forces. It states that instead of the military, he got a job at a Delaware shipyard.

Seems pretty clear that he was not in the military, so he wouldn't be in that photo.