Mage is a good generic term because it's gender neutral in English and doesn't overtly refer to any specific element or tradition that most people know about. However it does come from a genuine occult tradition, the Persian term magi, a Zoroastrian linkage, which is an interesting religion to look at. But most people don't know anything about Zoroastrianism so the term 'mage' works well as a neutral term in non-esoteric general fantasy works.
It's your story and world though, if you want to use witch, warlock, wizard, or mage, magi, magus, sorceror, sorceress, conjuror, or something else, and there are dozens of these words, it's your choice.
It's your story and world though, if you want to use witch, warlock, wizard, or mage, magi, magus, sorceror, sorceress, conjuror, or something else, and there are dozens of these words, it's your choice.