First, there is a fundamental difference between what is known as Magic, and what is known as Sorcery.
All demons have magic and all magic is demonic, but that does not mean that only demons can use magic. It is found naturally in demons and only demons, yet the potential for magic exists in everything, which is the reason why there are magi among humans. However, to use this potential requires it to be simultaneously destroyed as well as strengthened. Each time a caster (most often being human) uses magic from his own potential (from here on referred to as mana to help you draw a clearer picture), that mana is gone from them forever, but every time the magus destroys his mana, however, his body begins to produce more, naturally. Naturally. As I said, all magic is demonic. Though they are still not using the magic of the demons, these magi who produce mana naturally are on the path to paying the ultimate price, as activating this mana comes at a cost, most often being the magus' humanity. This is also why the Inquisitors do not use their mana. Instead, they draw their mana from the living environment around them, forcing the world to pay their cost instead.
The Three Types of Magic
Pure Magic
This is what demons and half-demons wield. It is, as it is named, the purest form of magic currently known and is generally destructive and deceitful in nature. The other two magic are nothing more than variations of this form. Full-fledged demons have more control over this form than those with diluted blood, as this form magic is tapped directly through their blood, allowing them to use it without any sacrifice. It is part of them, it is who they are.
Faustian Magic
This is the state that humanity can handle. As its name implies, a human who uses this magic essentially sells themselves to the demons, though this does not mean that they need to make a deal with demons to use magic. All this means is that since they lack the demon blood, to use the magic they must sacrifice something internally: their humanity. Every time a magus uses his magical abilities, he kills a small portion of himself as the cost for activating his mana. The larger the spell, the larger the damage to his soul. If he passes a certain point, the magus ceases to be human. Indeed, he is little more than a demon in a human shell. His mana limit might be at the highest it could possibly be at, but for all extents and purposes, the human he once was is dead. The nature of this magic is generally related to chaos and death, though since it is a form of Pure Magic, it does also grant them quite the ability for destruction.
Holy Magic
This is the Trinity’s magic. In many ways, it is more similar to Faustian Magic than it is to Pure Magic, despite visually appearing to be closer to the other. Holy Magic, to the ignorant everyday man, appears to not need a sacrifice, but this is far from the truth. Indeed, Holy Magic needs every bit as much a sacrifice as Faustian Magic. The difference, however, is where the sacrifice is taken from. The Faustian Magic requires the human to sacrifice themselves, an internal sacrifice, while the Holy Magic instead demands an external sacrifice. Each time the Holy Magic is used, the sacrifice is taken from a world. It does not have to be the one the Inquisitor is residing in, and indeed, only the newest of Inquisitors ever draw the sacrifice from their own world. This magic tends to become the Order to the demonic destruction and the faustian chaos. It is not the magic of life, but it is the magic of truth, and as I said, order.
Sorcery is a fourth, and rarest, "type" of magic, the mutated form, and generally only the most powerful magi can cast it. Like magic, sorcery can allow its wielder to accomplish nearly impossible feats that a normal being is unable to do, however that's generally where the similarities end. Unlike magic, sorcery is not demonic in nature, despite being stemmed from demonic origins. For those who can cast it, sorcery has no cost for its activation. In essence, Sorcery is purified magic, to the point of where it is no longer harmful to its users, though nearly impossible to create itself. For the average magi, Sorcery is magic, but not magic at the same time.