Semi-smokeless powder was made by the King Powder Company, which shared a common location with the Peters Cartridge Co. factory in Kings Mills, Ohio in the early days. I have never read anything authoritative on exactly what King Semi-smokeless powder (or Lesmok) was, except it has been presumed to be nitrocellulose incorporated with black powder or black powder ingredients. Some references indicate the King product went on the market first in 1897, while Lesmok came later. Lesmok was presumed to be ballistically similar and more or less interchangeable with Semi-smokeless, mainly, if not entirely, in revolver and shotshell loads. Peters supposedly replaced black powder with the King SS powder for their older BP loadings. I have not seen a reference to who actually manufactured Lesmok - maybe it came from King and was simply their same product with another name, or more likely, was made by duPont. In any event, I doubt it was manufactured by Winchester as I’ve read that Remington and UMC also used Lesmok. Both were used in the manufacture of match-grade .22 RF cartridges. I’d like to know myself just what King Semi-smokeless powder consisted of, and if Lesmok powder was anything different.
Back in my boyhood, in the 1950s, King Semi-smokeless powder was still available, and was very popular among the sizeable muzzle loading clan in Southern Ohio where I grew up (which was the original home of the NMLRA back in the 1930s), even though its manufacture had ceased during the Great Depression. I even shot up a couple of cans of it myself, but unfortunately I did not save the cans.
I’d say your Peters .25 box appears to be from the general WWI era but could be a bit earlier. I have a color picture of boxed Peters cartridges from a 1914 catalog with essentially the same label design.