The true definition of a "Magnum" is that where firearms are concerned it's a marketing hype term, sort of like Express was.
Magnum was first applied to rifle cartridges in Britain around the 1880s/1890s, and the differences in performance really weren't that great. Normally just a bigger case (as the magnum wine bottle, from which the term was taken, is a bigger bottle) with more powder.
Magnum performance kicked up a notch when smokeless powder was introduced, but the dividing line still isn't as clear as Taylor's definition would make it seem, as some Magnum cartridges are hard pressed to hit the 2,500 fps. mark, while other cartridges not named Magnums will do it easily.
A lot of people (at least in the United States) tend to think of the Magnum as being cartridge based on an existing round, but with a longer case, more poweder space, and more power -- the .357 Magnum as an outgrowth of the .38 Spl., for example.
But even that's not definitive, as the .256 Magnum and the .41 Magnum didn't have clear predecessors, nor did the .300 H&H/.375 H&H Magnums -- they simply sprang up out of design studios.
One final note of interest... S&W owned the trademark on the term Magnum as it was applied to handgun cartridges. I think it's slipped into the public domain now.