I'm really getting confused with all the talk about a .45 Frankford cartridge and a .45 S&W Special cartridge. Do these things really exist?

So far as I know FA produced (at least post-1876) only the .45 S&W cartridge for the Schofield and SAA, plus the larger rimmed 1909 version for the Colt New Service. The ONLY purpose I have ever read for the larger diameter M1909 rim was that the narrower rimmed cartridges in .45 S&W and .45 Colt calibers did not provide the fully reliable extraction performance in a swing-out cylinder revolver that the Army wanted. It had nothing to do with not allowing their use in the SAA, as the 1909 cartridge COULD be used in it, but only three cartridges in alternating chambers. The first ammunition for the .45 autoloader by Frankford was a 10,000 round test lot made up in 1906 for pistol trials, and that cartridge was somewhat similar to the later-adopted M1911 .45 ACP cartridge, as it had a 230 grain FMJ bullet. Maybe that FA test lot is mistakenly being called the .45 Frankford - I don't know. If that round was used in the triple-lock S&W, it would have necessarily been especially chambered for it, and that does not make much sense (using a rimless cartridge in a revolver).

Regarding the ".45 S&W Special", what is it and what was it chambered in? What would have been the point of developing such a cartridge, with the .45 Colt and the .45 S&W already being in existence?

All I'm asking is for someone to provide proof that the .45 "Frankford" and/or the .45 S&W "Special" cartridges referred to actually existed, separately from the .45 Colt, .45 S&W, or any pre-M1911 .45 autopistol cartridge (such as the 1906 FA test lot).