Ulysses S Grant, F Scott Fitzgerald, JEB Stuart, P T Barnham, FDR/LBJ are just some of the thousands of examples of this- Truman and Grant installed the 'S' just to have an initial. Is there a reason why this developed? Were they generally known as such in their own lifetimes?
It distances themselves from other people with the name. There may be many other Harry Trumans in the world but not that many Harry S. Truman. Media outlets especially do it with notorious killers (John Wilkes Booth, Lee Harvey Oswald, James Earl Ray) so that people with names like John Booth don't get confused with the guy who killed the president.