To my mind, Magic is essentially, the codified explanation for all of the things in any given setting that couldn't exist in the real world, its generally a 'thing' with many points of contact and many dimensions to it that have been designed to create a certain kind of world-feel and implications. The design might be explicit, like in hard magic systems, or it might be implied as is usually the case with soft magic systems. People can use it (spells, magic items) but it also exists naturally in the world (something like a dragon's naturally abilities, for instance.)
It serves to heighten one's suspension of disbelief by lubricating the impossibility of the stories with a 'rational' explanation that operates to create cause and effect, without worrying too hard about believe-ability (contrast with sci-fi, which traditionally makes small changes to the laws of physics, but tries to operate within real world ideas.)
Whats weird and interesting, is that we have 'magic' and 'Magic' lower case is things that are can be identified descriptively as magic because they exist in that 'rational explanation for the irrational' space of holding up the fantasy elements, but upper case can also be used to refer to a specific kind of magic in a world with 'competing magic systems' like psionics, which are totally magic, but aren't Magic, simply to show that its a different, competing system.
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