Every one of Ai-chan's stories use different magic systems.
In Seven Years of Hope, you can't cast any magic that you were not granted with. The protagonist had a choice of ice magic, fire magic or blood magic, though the canon is ice magic, granted by her adopted brother who was a descendant of King Frost. Herself has a divine summoning magic, where she uses her songs and dances to summon an avatar of the gods or an aspect of the gods.
In Hero Prince(ss), everyone has the potential to use any magic they trained in. But most people has an affinity to only one or two of Fire, Water, Earth, Wind. There's also Non-elemental magics. Divine magic is similar to Chaos magic, except divine magic is about order while chaos magic is about disorder. The protagonist acquired Chaos Magic after defeating the fallen angel at the demon king's castle and stealing the angel's divine sword.
In Age of Gods Online, magic is such a pain in the ass that people don't like using it. You first summon the element, then give the request, then provide the details, then give the order. It's much faster to throw a sword at people but otherwise, anyone can use any magic if they have the authority. The Mycrans can use Fire Level 1.
In The Hounds of Hell, magic uses a person's stamina and the types of magic one can use is determined by what ring that person wears. Gold rings provide abstract magic that mostly affect the mind or more utility in nature, such as Radar or Telepathy. Bronze rings provide magic that directly assists in survival and living, such as Accelerated Plant Growth. Titanium rings provide magic that has direct benefit in battles such as Fireball or Enhanced Strength.