There are a number of cartridges whose original designation included (and sometimes still does) the name of the company that developed them but is often dropped today, and the .44 Special is one of these. Smith & Wesson developed the .44 S&W Special by lengthening the .44 Russian case, and renaming it to include the S&W name, which is rarely used today. Subsequently, Remington worked with S&W to develop the .44 Magnum by lengthening the .44 Special case, and so the full original name of the .44 Magnum cartridge is .44 Remington Magnum, again often shortened to just .44 Magnum.
It's much the same story with the .38 Special, almost never referred to (except on S&W barrels, where it creates confusion for some with the different .38 S&W cartridge) by its original name of .38 S&W Special, the .357 S&W Magnum, and the .357 Remington Maximum, the latter of course never achieving the popularity of the .38 Special and .357 Magnum.
The situation with the .380 ACP is a bit different, because those alternate names are the European designations, i.e., 9mm Kurz (German for "short"), or 9mm Corto (Italian for "short"), along with some others that don't include the word for "short."
ETA: I see that BUFF provided much of this info while I was typing.