So i wikied thaumaturge to find that it does not have almost anything to do with ice so i decided to wiki all the rest of the elements too.
here we go:
(Ice) Thaumaturge - Someone who practices the art of miracles, or magic. When used in greek terms its refurring to more spiratual powers. Many saints were often called thaumaturge's.
(Fire) Pyromancer - Someone who practices the work of divinatation (similar to seeing your fate or looking into the future) through the use of fire and usually some other item or aspect. Examples would be of oracles burning bones in acient china to see the emporers fate. Though misused in most video games, the correct word for someone who controls fire is pyrokinesis.
(Storm) Diviner - Someone who practices the art of looking into the future, or predicting out comes based on different beleifs. such as burning bones and looking at the cracks, or looking at the shape of a cows liver. The azteces were diviners and they beleived their gods to be too.
(Death) Necromancer - Someone who practices the art of raising the dead in full body form or just spirit form. The idea was to use the spirits as oracles and ask them questions of the future or just plain wisdom. Necromancy was considered by some as Thaumaturgy for the miracle of bringing back the dead. others veiwed it as demonology and was forbidden. Now days it has a looser meaning, mostly connected to anything spiritual, such as a person trying to contact and an angel would necromancy.
(Life) Theurgists - Someone who practices the art in animating statues, with the idea of pleasing gods, or sometimes summoning the gods wisdom. It can also be seen as becoming your perfect self and becoming one with nature and the gods. Sometimes it means someone who looks through nature for divinitation or looking into the future.
(Myth) Conjurer - Someone who who practices the art of summoning Gods, Demons, Angels, and super natural agents. They usually would summon them for various reasons giving the spirits orders, most religons and cultures veiw conjuration as unholy or evil, but there are a few that rely or beleive in conjuration.
(Balance) Sorcerer - Someone who practices the art in all-around magic. They can go by many names for a sorcerer is someone who just practices magic. they are sometimes called magus (Note:thats the name of a lvl 30 rank wizard) wizard, mages,Theurmagist and so on.
so as you can see most of them fit in a certain way but a few are a bit off. it was just interesting to research all these.
I am impressed, though i wonder why Wizard101 used these words to name the practice of the school, when some of them don't even mean a thing about them.