My impression is that rune magic derives from writing in general having a magical quality. Dating from a time when only an educated, and hence special, minority could read and write. Magic seems to lose it's "magical" quality when nearly everyone can do it.
Magic does work best when it's got something to contrast against (typically the world-at-large being non-magical).
For me, whether something is magic or not depends on the setting and on how it's presented within that, handily most settings label magic as such so that it's easy for the viewer/reader. What magic is and how it works will vary wildly according to the setting that it's placed in, from the Science as Mathmatics of Charles Stross's Laundry Series, the Newtonian angle of Ben Aaronovitch's Rivers of London series or the magic-is-magic type traditional fantasy. So long as it all makes sense in universe.
Ideally magic should be rare and out of the ordinary - If it's everyday and humdrum then it isn't magic, in the same way that (to paraphrase Syndrome) If everyone's super, then no-one is. There doesn't need to be a set, single definition!