I would argue that the "pop cultural" idea of magic that you describe, the Tolkien and Rowling magic, is not something that was "introduced" to society. Rather, it is the remnant of centuries old folkloric beliefs about magic, mythology, and witchcraft. These tropes have obviously shifted and mutated over time, but they have been present since the ancient Greeks and Romans used to write about magic, alchemy, and other magical practices.

Rowling and Tolkien were certainly creative, but their creativity is not because of pure invention. They did not merely invent this idea of magic for their novels. Instead, they took storytelling tropes and popularly held beliefs about magic (many of them medieval) and infused them with interesting stories of their own.

The greatest evidence for this is in the background of Tolkien and Rowling themselves: J.R.R. Tolkien was a linguist who worked with such classic texts as Beowulf, the Volsung saga, the Nibelungenlied, the Kalevala, and the Edda. Many of his characters, themes, and descriptions of magic come directly from these earlier traditions.

Rowling, similarly, has a degree in classics. The Harry Potter series is littered with references to classical mythology (Remus, Fluffy the three-headed dog, etc.) and much of her magical culture comes directly from the Middle Ages: pointed hats, magic wands, Nicolas Flamel, alchemy, and the philosopher's stone, the first years' journey over the Lake turning the medieval belief that witches can't cross water on its head.

The ideas of magic and witchcraft have now become integral to fantasy and pop culture, but these tropes are nothing new and spring from much older traditions in which witchcraft and magic were believed to be very real.