Some people use the term "magnum," in the context of firearm's cartridges, to describe a cartridge modified from a parent cartridge with increased power or potency such as the 357 Magnum and 44 Magnum. In another case, the 41 Magnum had no direct parent cartridge but Remington wanted it to be characterized among the other 'Magnum' cartridges because its ballistics were similar. Some people give further distinction to cartridges that are magnumized version of magnums, referring to them as "super magnums." .327 Magnum, a magnum cartridge based on the .32 H&R Magnum is a "super magnum." Other such cartridges have been named as the .357 "Maximum."
There are cartridges that have been loaded to a higher pressure and power level without modifying the case length or other features from their predecessor. The .38/44 is one such cartridge, as is the 38 Super. Although both these were introduced prior to the 357 Magnum, they are not considered magnums, possibly just because the people that conceived them hadn't thought of using that term. Indeed the 357 Magnum is widely regarded as the first magnum handgun cartridge, basically because it was the first time anybody called a handgun cartridge by a name that included "Magnum" in it.
Was the term "magnum" applied to other firearm cartridges before the 357 Magnum? It was applied to the .375 Holland & Holland Magnum in 1912. Prior to this, I'm not aware of any earlier cartridges with "magnum" in their name. Why did H&H call it "Magnum?" Maybe because it sounded better than "Veloplex?" This apparently first magnum wasn't merely a slightly stretched version of some parent cartridge. The brass is considered unique. However, the .400/375 Belted Nitro Express did precede it and featured things like the case belt. The Magnum was the same caliber but a much larger case altogether. It was loaded to a relatively low pressure with cordite but gave excellent performance because of the enormous case capacity. Many future magnum rifle cartridges would be based on it. Weatherby introduced a series of Weatherby Magnums, which Winchester produced to Roy Weatherby's specifications. Winchester then got into making their own versions of Magnum rifle cartridges similarly based on shortened .375 H&H cases, but starting with smaller calibers. Remington similarly introduced an H&H-case-based Magnum in 7mm which Winchester had neglected. Later Remington introduced Ultra Magnums, which may differ a little from Super Magnums because the cases are based ultimately on the .404 Jeffrey rather than the 375 H&H. Winchester went on to introduce Short Magnums and Super Short Magnums which don't make a parent case longer, but instead deliver "magnumness" from a shorter or super-shorter case.
In the case of shotgun cartidges, magnums seem to be elongated cartridges of the same gauge, typically a result of stretching a 2.75" cartridge to 3". Apparently, stretching a 2.5" 12 gauge or 20 gauge case to 2.75" did not qualify it as a magnum. Stretching the 12 gauge cartridge to 3.5" resulted in a super magnum that is never called that but instead usually just 3 1/2", but sometimes "Magnum" or "Ultra Magnum."
As we can see, the process of magnumfication might involve increasing case length, pressure, or a larger overall case with the same caliber as some predecessor or legacy cartridge. In other cases, however, it may involve shortening a case to fit in a shorter action, keeping a low pressure to ensure extraction when hunting dangerous game, and even making cases smaller if pressure can be increased and powders more modern than cordite can be used. In all cases, magnumizing involves increasing the awesomeness.