In a sense, yes on both fronts. A long time ago, in the olden days, belief in magic was very strong. So magic came to be. From that belief, a great tree, Yggdrasil bloomed, and it granted power to everyone who would use this magic. Men and women alike. Witches are the most well-known, of course, but plenty of men have picked up a wand in their time.

[ ... But... ]

... Over time, though, Yggdrasil began to die out. And the roots of its power that connected the entire world... were all that was left of it. These are what we call the Ley Lines today, and we use them for travel from one place to another, but we also use them for magic. It's the energy from the Ley Lines that allow a witch to do magic. Without that, no spell will work.

... But the Shiny Rod is different. It's from the old world. It can perform magic without need of any source nearby.