Correct. The .38 Long Colt was the Army's service cartridge c. 1898-1905, and was the basis for developing the .38 Special cartridge (same rim diameter, body diameter, etc, simply lengthened to contain a larger powder charge and morphing into the .38 Special), reportedly due to reports of poor terminal performance of the .38 Long Colt in combat against Moro tribesmen in the Phillipine Insurrection.

The .38 S&W case has a larger rim diameter and larger body diameter, and many (most?) revolvers chambered for this round required a larger diameter bullet as well. A revolver chambered for the .38 S&W cartridge may, or may not, accept and fire .38 Long Colt or .38 Short Colt cartridges, but the reverse is much more doubtful.

Back in the latter 19th Century the major American firearms manufacturers made weapons for proprietary cartridges of their own design and development, frequently also running ammunition manufacturing plants to produce their own products.

The .38 Special cartridge appeared in the 1905 S&W Military & Police revolver, presented for military trials c. 1905. The Army Ordnance Board decided to pursue a .45 caliber pistol instead, resulting in the Colt Model 1905 which was rejected for lack of positive safety devices, and performance of the 200-grain ammunition developed for it. Colt responded with the Model of 1911 US Army pistol with grip safety, thumb safety, disconnector safety, and half-cock "safety", along with a redesigned cartridge with 230-grain bullet, and the rest is history.

The Model 1905 S&W M&P revolver went on to wide acceptance and use by law enforcement and military units around the world, and was further developed into dozens of variations with many still in regular production and use.

The .38 S&W Special cartridge became one of the most popular handgun chamberings in history, with many millions produced by dozens of makers around the world. Smith & Wesson apparently never troubled themselves needlessly about the fact that this development was originally based on a Colt proprietary cartridge dating back to about 1870, when Colt was busy converting thousands of percussion 1851 Navy revolvers to cartridge guns.