Does this make sense? Witches and warlocks (who are people born with hereditary magical power) classify magic by what element symbolises it best. The elements of magic are as follows. A few examples for each are as follows. Fire: Purfication and destruction are fire arts, as is crafting things. Water: Healing, as well as changeing things are under the auspice of water. Metal: Telepathy and empathy are popular metal magiks. Wind: Famous wind magic includes flight and teleportation. Earth: Shapeshifting is of earth. Lightning: Necromancy and giving life to things that've never had it are magical powers belonging to lightning. Any more possible elements? And where might other famous magical abilities from fantasy?
Anything weird that is under a classification could be used as an example. I'm not much into pokemon, but it seems like they mainly follow elements, plus having ghost and grass. In another game I play, there were element-like spells, but one was sound. Final Fantasy Blue Mages actually sound kinda weird, and then there are the Dragon Aspects in World of Warcraft. I wonder what a technologically-talented mage would be. If I can magically break computers, I imagine there is someone who can magically manipulate them. I think you're getting distracted with world-building, just start writing the story and let them do magic as they will. If there is any moment where they could magic themselves out of a situation, come up with some excuse why they need to be a bit more creative. After your rough draft, see if the rules occur to you, or if you even need to worry about it. I once came up with a type of sorcerer that got nicknamed "Dollmage." Apparently they were common enough that there could be several in a small town, but they were rare enough that they were pretty much unknown. (Kinda like how the Soul Reaver fans here seem surprised that there are others.) I didn't have the rules at first, except that it was instinctual. Even I didn't know the reason for the limits on their powers until I re-read it five years after writing it.
Your breakdown makes sense, but realize that magic is whatever you make it. This is your tale, you can turn the world into whatever you wish it to be.
A techonology focused mage, or Technomage, would probably involved multiple elements rather than a single one. Take a steam powered engine for example. That would probably incorporate elements of water, wind, and a bit of fire. Yes, we call them hackers.
