Disclaimer: massive wall of text. Read at your own peril. I hope the pictures make it more bearable, though.
My current understanding on how magic casting works is that mages draw in mana from the atmosphere into their bodies, thereby creating personal magic energy. The limits of one’s personal magic energy holding capacity is what is represented by the magic bar in AA, WoM and AO, and can be increased by training and using items (though the latter can just be assumed to be a quirk of the games not representative of the actual lore). This is suggested by Vetex in this thread:
The shape/qualities which this magical energy can manifest in are dictated by one’s mind. It is the mind that allows mages to cast arcane spells, and it is by the use of these minds that most great wizards have made their mark on the world. Most spells, however, are cast by the use of magic circles, which most likely function as magic programming units. I make this conclusion based on three key pieces of evidence:
Magic circles can be detached from the user’s body,
Elemental magic energy can be infused into auxiliary devices without mana, and
The personal magic energy of a magic user appears to take on the qualities of their minds while they are drawing in mana from the atmosphere.
This implies that magic circles are neither responsible for the emission of personal magic energy nor for the conversion of personal magic energy from atmospheric mana to arcane magic. This also weakly implies that the source of the debilitating effects of primitive magic in the Golden Age of Magic were caused by the cost of controlling personal magic energy, and possibly mana itself. Since we know that the casting of primitive magic somehow involves the soul, this further implies that the soul is also capable of manipulating mana, which might be a hint as to the composition of magic circles.
Assumed flow of mana within the body. Note that arcane energy can be converted into spells via either soul energy or magic circles.
There is a bit of a grey area, however, in the way in which the mind converts personal magic energy to arcane magic, or in whether personal magic energy and arcane magic are distinct at all. For the sake of clarity, however, I have assumed that personal magic energy is separate from arcane magic, as the graphic above displayed.
Edit #3 starts here
Well, initially I had assumed that this was an unsupportable position to hold, but that might not be the case. As pointed out by @Jar, users with multiple minds exist, one of which is the Player in Arcane Adventures. The action of selecting which arcane element to use in a spell seems not to be reflexive but intentional, suggesting that magic users can manually choose to activate specific minds.
We can reliably assume that it is impossible for a single unit of mana (whatever that might be) to possess the natures of both minds at the same time. This assumption stems not from any direct evidence but from the total lack of evidence to the contrary, as well as the similarities that such a system would have from the mutation mechanism in Arcane Adventures and the needless complications that such an addition would add to the magic system.
From the two assertions above, we derive two conclusions:
The conversion from personal magic energy to arcane energy is manual, not automatic, and
It is possible for a magic user to hold personal magic energy without converting it to arcane energy
Although the second conclusion arguably does not follow from the assertions made (after all, it is possible that mages must immediately convert any drawn-in atmospheric mana to arcane energy of a quality dictated by one of their minds), but this seems more contrived and thus significantly less likely than the conclusion made above.
In other words, the use of atmospheric mana by magic users is dictated by this decision tree:
The decision tree does not consider the possibility that a magic user could absorb atmospheric mana and then convert only a portion of the now available personal magic energy into arcane magic, though there is no reason to assume that this is impossible. Similarly, the possibility of holding individual reserves of multiple kinds of arcane energy at the same time is not conveyed.
To explore more the possibility of holding multiple reserves of arcane energy at the same time, however, one would need to first assume that the way in which multiple minds are represented in the games is slighty inaccurate, as switching from one mind to another is suggested to be as easy as converting all the arcane energy in one’s body into another kind instantly. This slight inaccuracy can be explained away by reasoning that forcing the player to have to charge again after switching magics would be inconvenient.
Edit #3 ends here
You might have noted that I referred to the spells commonly used by the player in AA, AO and WoM as arcane spells and not simply spells. I think the disambiguation is necessary, as other classes of spells exist, and there is yet no evidence that these spells require magic circles of any kind. Even if we are to assume that only souls and magic circles (if they are indeed distinct) are capable of manipulating mana, a person capable of somehow storing soul energy would still be theoretically capable of magic, as far as we know. Perhaps it is by this indirect method that the witches in AA and WoM cast spells on objects. Maybe the ingredients are vessels that contain easily extracted soul energy, or qualities comparable to magic circles.
The mechanism or organ responsible for the drawing in of atmospheric mana is not directly stated, but, by simple process of elimination, it can only be one of three choices:
The soul, or
Some physical organ like the lungs.
Any other possible mechanisms that might be held responsible for mana in-drawing would risk over complicating the magic system, and would be yet another entity that Vetex would need to come up with an explanation for; both undesirable positions to be in.
Although the last proposition is admittedly a bit of a stretch, the first two are perfectly tenable, though with very different implications. If the drawing in of mana is accomplished by the soul, then non-mages are capable of it as well. This is completely useless in a fight as mana in its rawest form is completely benign (explored below), but it does mean that a non-mage is capable of using other non-mind involved magical arts that might exist in the Arcane Universe. Of course, if the drawing in of mana is accomplished by the mind, then these possibilities are shattered instantly. The direction that Vetex takes (or might have already taken) will depend on how much supremacy he wants wizards to have in the AU.
I personally believe that it is the soul, not the mind, that is responsible for mana in-drawing. This is because in the lore, it is revealed that it is possible for humans to grow tired physically by using magic spells. Since the mind is physically located in the brain (can’t remember the source for this information, so you’ll have to trust me on this one), it’s more likely that the soul is responsible, as it is easier to imagine one getting tired from soul energy overexertion. This isn’t solid proof - mentally taxing exercises can be quite taxing, after all - but it does favour one theory over the other.
Since alchemy evidently does not require magic circles, there are only two possible ways in which they might be cast:
By the use of soul energy, whether from themselves directly or from existing materials with soul energy, or
By the use of some other material or phenomenon capable of programming spells.
Both theories are equally tenable. The first can be justified easily: since the soul energy extracted from alchemic ingredients is not as strong as that in humans, and since it cannot be commanded to control mana in the desired manner directly, then different soul-bearing materials, all varying in some quality within their souls, must be fused together in a very specific order and ratio in order to create the desired effect. Perhaps the difference between a human soul and the soul of any other organism is only in the control that humans are capable of exerting upon their own soul.
The second is justifiable simply by virtue of the AU being a fantastical universe. Vetex could very easily make up some other soul-bearing material capable of causing magic phenomena to occur, though this would mean that he would be creating yet another avenue by which non-mages might master magic. Even with alchemy, creating automatically alchemizing machines is already a possibility for non-mages, though perhaps not with the technology present in the AU.
Magic energy is most likely completely divorced from the physical realm and cannot truly interact with terrestrial entities, only appearing to do so when interacting with the magic energy (whether personal magic energy or arcane energy) within another terrestrial organism. This is gotten from these two quotes from Vetex:
This is rather problematic. Firstly, we know from details in the lore and from the games that magic spells are capable of causing physical damage to terrestrial entities. It was in the battle between Theos and Durza that most of mankind was wiped out, and that the continents were scattered across the seas. These feats imply that objects can be affected by spells. It becomes less of a problem, however, if one concedes that atmospheric mana freely courses in and out of all terrestrial matter, and that any object completely devoid of it would have gained a rush of it within the explosion and would have subsequently been destroyed.
Secondly, it presents a large problem when dealing with elemental curses, especially those of the more physical nature and even more so when dealing with extremely powerful mages. If humans are capable of actively forcing atmospheric mana into their bodies, then the most powerful of them must be capable of stopping atmospheric mana from leaking into their body by directing the absorption force away from themselves. This is almost definitely possible, as the long ranged spells in WoM are accomplished by emitting either personal magic energy or atmospheric mana towards a non-local target.
A placed explosion type Ink spell, World of Magic
This isn’t much of a problem when dealing with non-curse users as it leaves them vulnerable to any environmental attacks like falling rubble, but, as curse users physically become their element, they have the advantage of essentially being able to cast magic spells (or their nearest equivalent) while being impervious to magic attacks.
Of course, this is all predicated on the assumption that their opponent can’t simply hit them with an attack that has a stronger magic force vector than the force they are devoting to repelling atmospheric mana, but I assume that such a feat requires that both wizards be at least on par with each other in magic strength, and not even a little bit less. This is still a massive advantage though: by using this technique, the curse user does not incur any loss for getting hit by an attack stronger than their maximum strength and can simply tank most attacks the opponent flings at them, while the opponent must always dodge, counter, or block their own technique at the cost of their resources. The only thing the curse user has to fear is the physical toll that constantly using this technique will have on them.
Assuming I am right and curses do carry this massive combat advantage, it would go even further to explaining why curses are so heavily valued, even with the existence of magic. Although one might assume that a fire curse user would be at a disadvantage with a water mage of the same level, the ability to effectively make the non-curse user work harder for less will eventually lead to victory for the fire curse user almost all the time. They would have to be challenged by a user significantly stronger than them to be defeated.
As pointed out by @a_nerd, there are actually examples of animals that possess magical qualities in the lore. This would suggest then that human souls are not distinct from all other souls in a dichotomous way, but are a part of a spectrum of which they are presumably on the extreme end of soul malleability (in this case, this refers to the ease and variability with which the soul can issue commands to magic energy).
This is also further supported by the fact that magical acuity varies greatly between lineages and even more so between different sentient races. Skylians, for instance, probably have souls more malleable than typical humans.
I meant terrestrial entity here.
@Jar brought up the existence of multiple minds, which I had not considered in my theory. That prompted me to add this paragraph.
Silly mistakes. Pointed out by @GlitchingEclipse.