I put together a little primer on the supernatural elements that our protagonists would know in character, in case this assists anyone in making a decision. If there's no change when I come to write the next update, I'm going to declare myself a tiebreaker and go for the choice I think is more interesting to write.
If you think you recognize something: yes, you probably do. I tried to make interesting twists on the magic for this setting, but it's definitely drawing inspirations shamelessly.
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For mortals, there are generally accepted to be a few specific methods to acquire superhuman power. There are enlightened martial arts and the five paths of magic: Conceptual Magic, Elemental Magic, Contracts, Tome Magic, and Divination. It's far from unheard of for any given great man or woman to take an eclectic approach to these, mixing and combining various talents to find the best tricks that the individual can achieve. Thus, terminology is quite inconsistent, both due to the radical differences possible and the fact that, even with the seer network, long-distance communication in the Shallow Ocean is slow and unreliable. The most typical terms are just "hero" and "mage". A hero is someone who has more combat power than average when they fight, either with a weapon or barehanded: typically, anyone who could defeat at least three normal fighters at once is going to be called 'hero' by someone. Mages are those whose primary talent is magic.
Not every person who can use magic is considered a mage. There are countless people with some small talent, usually some weakly-bound concept or a specific effect or two that mimics tome casting. From the village elder who has some power to soothe pain to the champion pearl diver who can charm oysters into giving up themselves to the hedge witch who can bless or wither crops (a bit), these hardly count. For one, it doesn't put them beyond human capacity, for all that their specific trick gives them some new ability. Something more is needed.
The enlightened martial arts are a varied lot. Most established schools expose some intrinsic flaw in how the world works, and exploit it ruthlessly and in increasingly difficult and subtle ways, as mastery grows. Most can be used either bare-handed or with weapons, though many clearly lean to some specific angle. A comprehensive list would be difficult and beyond what you currently know, anyway. There are three that you know of that are currently in use in the capital. The first is Bear's Mantle, focused on superhuman strength and shifting weight in impossible ways to make oversized weapons both easier to handle and hit even harder than they should. The second is Viper's Kiss, which manipulates the user's own body chemistry to generate poisons they can pass with a touch, as well as synthesize other things to boost their own abilities. The third is Untouchable Blade Style, which apparently is creepy to watch and is great for one-on-one fights against other people, but you don't know anything about its mechanisms. There are a few others you're familiar with. Stroke of Midnight is a technique that combines divination magic to fight both in the present moment and prep for the future, while gracefully avoiding attacks and hazards, though its users occasionally get so caught up in the future that they falter in the now. Screaming Fiery Monkey involves acrobatics and converting body heat into flame attacks. Swirling Winds reads the air around and uses small changes, caused by its user's own breath, to control thrown and missile weapons to attack beyond sensible range, redirect in midair, or just fire absurdly quickly.
Conceptual magic is probably the most common type, and is considered close to the intrinsic supernatural power of a god, angel, or spirit. By binding a specific concept to the soul through meditation and enlightenment, the user can focus will to evoke related effects. For instance, someone who has bound the concept of "Fire" can do things from lighting a candle to throwing great balls of fire (at least, if the user has enough magical strength and stamina), but the more they press themselves, the more they can pull off effects tangentially related to the concept, such as sparking fiery anger. The concepts themselves aren't necessarily perfectly described by one word, as they are concepts, but it's rare to find. Concepts can also change over time, based on usage and intent. Wind and Music are somewhat infamous for flipping from one to the other, but many concepts somewhat overlap each other, meaning that it can take some observation to figure out the exact concept of a mage you don't know. Other common concepts to bind include water, weather, fields, healing, earth, and shadow. A powerful mage who binds an unexpectedly useful concept is a staple of many tales, but rare in reality.
Elemental magic is the most physically oriented magic. Elementalists, as they are often called, have to step out into a dangerous situation where two or more elements meet, and find an internal balance between the two. This lets them control and shape both the related elements, though unlike concept magic, it doesn't extend to more metaphorical connotations. Still, someone who is properly open to it while walking along a beach during a thunderstorm may find that commanding the waves and summoning lightning has great utility for him. Poison elementalists are always some of the most sought-after, because of their ability to extend a poison to food stores, to protect it from vermin, then counter it when it's time to eat. Although the strength of the effect depends on the elementalist in question, much of the reshaping of the ocean and islands to make them friendlier to mortal lives, as well the speeding of ships along routes that might not be quite in line with natural currents, spin from the actions of teams of elementalists. If not for the danger involved in gaining elemental magic, it would be even more popular.
Contracts are a form of magic that exploit the differences between mortals and spirits. The mortal gains some powers related to the spirit they contracted with, and the spirit can extract whatever concessions both agree to. Contracts are absolute to the parties involved: the magic structure is delicate. If the mortal agrees to a blood sacrifice every moon, the magic cannot consider extenuating circumstances that weren't hashed out ahead of time. Far-sighted or experienced spirits thus tend to offer contracts with detailed clauses to cover minor or unintentional breaches. Contract powers can be both broader and more esoteric than conceptual magic. Between this and the relative simplicity of gaining contract magic, contract magic is extremely popular. Beyond the delicate structure, another downside of contract magic is that the spirit feels, in the back of their mind, something of the mortal's thoughts. This is not a deep mind-reading ability, and is often mostly a headache, both for the mortal with minimal privacy and the spirit who may not want to listen in or who may get distracted from too many too disparate voices and suffer various mental maladies from it. To combat this, most of the spirits (such as gods) who have a large number of contractors specify codes of conduct for the 'clerics' or 'paladins' who echo their power. A god who values cold logic and even tempers may demand celibacy, or something close to it, while a deity of passion and desire may demand, ahem, something else.
Tome magic is the most difficult branch to master, but offers the broadest possibilities. It approaches magic from a studious, logical perspective, where known theories lead to specific results. Magic on this level is intricate and arcane, however: a spell to light a candle and one to throw a fire ball may have almost nothing in common, and you certainly can't replace the one with the other. Still, nothing prevents a tome mage from knowing a wild collection of spells: something to silence movement, to hurl bolts of force, to read the lips of someone facing away from you, to generate giant sticky ropes. Tome magic is called as such due to how complex spells are: without a substrate to simplify things that other branches allow, tome mages write down most of the spell and connect to it when they cast. Usually, this takes the form of a book, hence the name, where the caster touches the inscribed magical symbols and makes it part of the spell, only finishing the spell in their mind to decide such things as "where do I aim this" and similar effects that are best decided on the fly. Some casters will use other things to hide the powers they have with them: staves with subtle magic circles or perhaps clothing with the spell shape stitched into the cloth. Sometimes, a spell is learned so thoroughly that no external aid is needed, which is usually the case for those who pick up one or two spells that are technically "tome magic". For them, without further theoretical grounding or broader applications, it ends up as singular tricks. For a lot of non-mages, they think of tome magic as the highest or best type, assuming that its challenging nature and broad ability completely counter its lack of flexibility. In fact, few mages dedicate themselves solely to tome magic, and most have at least one bound concept or some elemental ability.
Tome magic is also closely related, fairly intuitively, to the creation of magic items. Laying down spells logically and using physical aids is only a skip away from making the entire magic item. Technically, most magic items are a type of artificial soul, though not one that has any sapience or sentience. Instead, they create, for instance, a soul in a sword that exists only to cast "my blade is sharp, hard, and casts a glowing light" over and over, forever, thus creating the effect. Despite centuries of effort, no one has been able to create a soul with any more of a self than this. The most complex they've ever achieved is something like "cast my spell when this condition is met", such as the fancy lights for the rich that turn on or off when verbally commanded to. Potions are distinct from this; they use a special brewing process to arrange most of the spell, but not to attach it to anything. It simply stays in a specially-prepared glass container that doesn't react to it. When used, the soul that they use to 'cast' themselves is the one holding the container. This means that a potion spilled during an earthquake will be wasted, but will not trigger itself. It also means that mechanical aids to pour out a potion while the setter is far away tend to be frustratingly difficult to arrange, and tend to require a specialize magic item to get any effect.
The last of the paths of magic is divination. Divination is the only path that does not seem to be able to be taught or imparted. Mostly, it seems to be a gift people are either born with or not, and training simply enhances and helps this. Seers, as users of divination are often called, can typically see communicate with each other over long distances, if both parties are willing. This is the "seer network" that constituted most fast communication across the Shallow Ocean. Seers can also sometimes see distant locations, catch frustratingly almost-but-not-quite trustworthy images of the past or future, intuit people's thoughts and motivations, or otherwise manifest knowledge they shouldn't be able to have. The exact set of talents varies by individual.
