Ancient Magic is Often "Divine"
In reality, magic appears to have been mostly associated with divinities and divine powers. It was not until more recently in the history of Humanity (and western culture) that we think of doing magic without divine influence. This means your people would have a similar attitude as the ancient world had towards magic.
For instance, Voodoo appears to be reliant upon relations with spirits to perform magic. One can call these relationships "divine" and therefore this is "divine magic." This is true for most (if not all) ancient cultures; the practitioners of magic were those who are close to spirits or gods, and this relationship allowed them to perform magic.
The wikipedia article on Magic in the Greco-Roman World has some interesting insight into the difference between "arcane" and "divine" magics. The arcane, according to this discussion, involves tricking gods or spirits, instead of supplication to those spirits or gods. D&D players would generally label all of these magic-users as "divine," as the actual source of the magic comes from divine sources. (More on D&D later!)
Astrologers and other fortune-tellers may be the closest thing we have to ancient magic users who do not need direct connections with divinity to perform their work. Of course, astrologers would read the stars, and it would be much less about exercising power as simply being trained to read those. The same argument could be applied to many types of fortune-telling; that it isn't magic, but simply being well versed in drawing conclusions from natural processes.
Vs. Other "Traditional" Magic Systems?
First off, Gandalf is more like a demi-angel, so the magic he uses is therefore divine. Sauron's magics would also be seen as divine, although divine in the sense of "above mortal power," as he is on the same level of "divinity" as Gandalf. The magic, or sorcery, of the elves seems to be not divine, but may be seen as divine gifts. Your culture of magic use may be seen as more religious than in Tolkien's Middle Earth, but could easily have a similar setup. It depends on how codified and formal your religions are.
Even Merlin, from the tales of Arthurian Legend, supposedly has some divine blood in him. In some renditions, he was born from a succubus and a mortal man, with the succubus part of his heritage giving him magic. While this is could be seen as a basis for "individuals casting magic without divine help," the fact that a succubus was involved could make his magic fall into the "divine" magic camp. Once again, your magic system would be different than the magic seen in Arthurian Legends, because this divine influence is well established.
The advent of D&D may have helped cement the idea that you can use magic without the need of divine help. Once again, this may be due to figures like Merlin or astrologers, who may or may not have had an explanation for their magic. Obviously, such a notion in this world your making is an impossibility, but perhaps one that people wish for. You may want to consider borrowing cultural elements from divine casters in D&D. Divine casters often acknowledge or at least nod to their patron deities.