There are dozens, scores, thousands of different magic systems in fiction and history. The latter, of course, vary by time, culture, and views of magic.
I recently read someone arguing that culture determines the magic system. This is true, to the extent of historical Earth magic systems because they were/are all based on how a given culture or subculture views the world and magic itself. However, the assertion fails, I think, when it is applied to fictional worlds where magic really works. Or perhaps that depends on how we define the phrase “magic system”: do we mean how magic works or how a culture limits and trains the wielders of magic?
The latter is clearly influenced by culture. This definition of magic system involves the artificial limits that a society places on what magicians are allowed to do or learn. It includes the laws enacted regarding the use of magic and its practitioners, usually to control those who possess an ability that the rest of society cannot understand. This definition also covers how one learns magic—self-study, apprenticeship, formal school, or whatnot. If magic is illegal, for instance, self-study and secret apprenticeship are the most likely means of learning. If magic is legal, but the non-magical government wishes to control it, then government controlled master-apprentice relationships or formal schools create a means of limiting magic. In the latter case, by regulating instructors and what it is that instructors can teach (we see an excellent example of this in J.K. Rowling’s HP and the Order of the Phoenix when Cornelius Fudge appoints Umbridge to Hogwarts as the government’s “inquisitor” and person on the inside).
However, if we use magic system to mean how magic works, which is the definition I typically use, then cultural influence is not an issue. Rather, in this definition magic influences and determines society. Why? Because this definition refers to the immutable, natural laws of magic. It treats magic like gravity, magnetism, and other natural forces. It determines where magic comes from. It explains where the energy to produce magic originates. If energy is necessary and expended, it explains how energy is tapped, used, and recovered (even if the effect is natural fatigue). The magic system determines whether the magician is physically, psychologically, emotionally, other otherwise drained from the act of casting spells. This definition reveals who is capable of learning and using magic (anyone who studies or only those with genetic talent, for instance). It governs the rules and natural laws of magic, e.g. what magic is and is not capable of. This definition also involves what acts are necessary to use magic—ex. force of will, rituals, hand gestures, words of power, sacrifices (of wealth, will, blood, self, etc.), and/or focus items (wands, staves, rings, amulets, etc.).
Michael Scott, Thirteen Hallows—Ritual or powerful artifacts necessary, blood sacrifice and sex can enhance raw power, it seems that anyone can learn but those with talent have an edge, learned by self-study
J.K. Rowling (Harry Potter series)—Wands and gestures are necessary, words of power are used initially, energy seems to come from outside the caster but concentration creates fatigue, only those with genetic talent can learn then talent comes into play, learned in schools, Dark Arts forbidden by society; multiple magics exist and follow different rules (Wizard magic, House Elf magic, Goblin magic, Centaur divination)
J.R.R. Tolkien (LotR)—A staff appears to be necessary, magic is only granted to five Wizards and, to some extent, one line of Men, words of power appear to be necessary, magic use by Wizards leaves a personal signature, cannot be learned; multiple magics exist with their own rules (Wizard magic, Elf magic, Dwarven magic [doors, forging], Ent magic, Sauron’s magic, Numenorean magic [Men])
Robert Asprin (MYTH series)—Anyone who studies can apparently learn, only takes study and force of will (visualization of effect), powered by natural magic field but can recharge by tapping ley lines, worlds without ley lines make magic difficult or impossible, both apprenticeship and schools are possible depending on the world
Fritz Leiber (Lanhkmar series)—Anyone can learn, but magic comes at a great cost that ultimately affects the magician’s appearance, magicians are secretive and feared (at least one is clearly based on Baba Yaga), magicians take on apprentices
Rick Riordan (Kane Chronicles)—Magic requires the bloodline of pharaohs, also requires the crook and flail, forbidden magic (merging with the gods) exists, usage drains the caster, words of power can be used (but are dangerous), learned by situational mix of apprenticeship and school, certain Egyptian artifacts allow transportation magic around the world