There is a lot of confusion about this, a lot of it from people who ought to know better.
The first cartridge is the 45 Colt adopted by the US Army circa 1873 for use in the Colt SSA.
The second cartridge is the 45 Smith & Wesson intended for the S&W Schofield revolver which the Army also adopted. This cartridge has a shorter case than the 45 Colt and a wider rim for the star extractor. (The 45 Colt is really semi-rimmed.) Only three of these at a time will chamber in the Colt SAA due to the wider rim.
The third cartridge is the 45 Short Colt. This is the 45 S&W with a narrow rim like the 45 Colt so that it may be used in both the SAA & the Schoefiled. This at the expense of a reduced load in the SAA and the risk of jamming under the extractor in the S&W.
The fourth cartridge is the 1909 Colt 45. The case is .010" longer than the 45 Colt, it has a wide rim like the 45 S&W and was adopted by the Army for use in the Colt New Service Model of 1909. It was loaded with a 300 grain bullet. At the time of its adoption it was viewed as a stop gap between the 38 Long Colt Army round and a 45 caliber self loader of some yet to be determined specification. The 1909 New Service is not chambered for the 45 Colt although it is intended to accept the cartridge in a pinch. It will shoot low using them. The 1909 cartridge will not chamber in a SAA with proper chamber diminsions. If the chamber is a bit long you can still only put three in due to the wide rim.
At this point experimental versions of the yet to be finalized and named 45 ACP appear. One of these was the 45 Luger using Frankford Arsenal bullets and DWM cases.
Then the 45 ACP was finalized and adopted by the US Army.
Next the 45 Auto Rimmed, a rimmed version of the 45 ACP for use in revolvers.
Then a 45 Winchester Magnum, a stretched version of the 45 ACP that never really caught on.
Last, but far from least, the 454 Casull.
There was never a cartridge properly called the 45 Long Colt although the term was first applied in the late 19th Century to distinguish it from the 45 Short Colt. (Not from the 45 S&W as every one supposes.)
There was also a 455 Colt cartridge, a civilan version of the military 455 Webley. It was once quite popular in Canada and occasionally seen in the US. Enough so that I bought a good many boxes of it for my Webley and my S&W. The Colt New Service and the S&W N frame were both chambered for this round, both for sale to the British military and for civilian sale in the US and Canada. The RCMP carried them at one time.
As a final note, when I go to the gun show and pick up a piece of modern brass headstamped 45 Schoefield, I find that it has been manufactured to 45 Short Colt diminsions with a narrow rim.