Witches are a cultural by-product of mostly-Western Christianity.
Umm no, we actually don't have anything to do with Christianity. We are a survival/revival of the old nature religion which existed previous to Christianity.
The origin of the word itself is unclear and there are several theories, dating variously to Old Norse, Middle German, Old English or Middle English. There is potentially a previous Proto Indo-European root that has been lost to memory.
In Wicca we generally call ourselves either Craft (short for Craft of the Wise) witches or Wiccan. Just to confuse things, not all witches are Wiccan nor do all Wiccans consider themselves witches, however most of us use the terms interchangeably.
Anyway, magi/magus/mage is another possible word, as you point out. Sorcerer, conjuror, enchanter, are also possibles, though technically each of those refers to a specific type of magic. Vitki is a Norse term (note the similarity to the word "witch"). Strega/stregaria refers to a specific group of witches from Italy. Bruja is the Spanish term. Mystic, wise woman/man or shaman are also viable.
I agree with others that calling someone a "caster" smacks of Dungeons and Dragons and would throw me out of immersion in about .3 seconds.
Perhaps it would help to know more about the specific type of magic your casters use, and as said, the origins of the group.
Also a pet peeve: Please o please don't call your folks "warlocks." The word means "oath breaker" and refers to someone who deliberately set out to infiltrate a coven and turn them over to the witch hunters.