"Witch" is not necessarily the cognate of "wizard" in any way. A wizard is someone who has developed some form of wisdom (wiz-dom). It comes from Olde English, as the root is "wis", meaning wise. The conversion of that into a personal noun by adding suffix -ard came later as noted below. Its use even then reflected at first and primarily the quality of wisdom as attributed to a person. It has come to mean someone, particularly a male, who has developed magical powers.When and how this happened I don't know.
A witch, on the other hand, originates in Olde English as well, and had both male and female forms, and implied a person who had magical power and this was its explicit meaning. It specifically meant someone with the power to bewitch others, usually of the opposite sex. The human equivalent of a demon of seduction, basically.
Male witches were called "wicca" and females "wicce", and the verb was "wiccian". If you pronounce the c's correctly, it sounds like a tch. So "witchah" and "witcheh". And if someone is "wiccian" you, they are "witchian" you, or "witchin" you. You are "bewitched".
In fact, witches were thought to be more conversant with spirits, and were thought even to have had congress, even sexual congress, with spirits. But perhaps it was more correct to have thought of them as Celtic shamans and shamanesses, and oracles later to have developed into institutionalized priesthoods, mostly Druidic.
So "witch" is far more magical in origin than "wizard", but wizardry has somehow come to be conflated with a notion of possession of magical power.
Magicians, being of a group called "Magians" and referred to as "Magi", were actually a sect of witch/wizards from the ancient Iranian plateau. The costumery often associated with "wizards" comes from that origin, and is reminiscent of a Scythian Magus, perhaps. They were scholars and mystagogues, and priests, not necessarily in that order.
Perhaps it was they, or rather Greek and later European interpretations of them as iconography for their own purposes which transformed "wizard" into something looking like a Persian Magus (or Mage, Magian, or Magician).
So in the end, for purposes of modern (not necessarily RPG/Fantasy Fiction) uses, we could say that witches represent a kind of innate magic with a propensity to charm or fascinate, wizards represent a profound intellect which is often the confluence of intelligence, intuition, experience, and discipline, whereas someone who managed to combine natural magical power with human mental adroitness would be some sort of psychospiritual renaissance being, a magician of sorts.
John Dee and Isaac Newton represent magicians in this sense, and are the modern representation of true Magi (in English terms). Zoroaster would have been a true Magus in his day, although the destruction of the libraries of Persepolis obscure what that would mean in detail, along with the rest of that tradition. Indeed, for all we know, much work was done there which is the forebear of modern science. Hard to be sure. But that's an aside.
Modern witches, in effect, would be female entertainers, and male entertainers, in any line of work or influences, using charm, often accentuated and augmented by psychotronics and other powerful technology. They're all over the place. Wizards are the deeper scientific and Machiavellian minds, working as either modern viziers, or ministers, giving tyrants advice, or in the former case, giving methods and technology to implement such advice. You have think tanks, councils, foundations, centers, and other "round table groups" just full of wizards and witches, acting as viziers for the ruling elite. Perhaps you have some evil magicians working in those folds as well, such as psychological operations officers high up in the intelligence communities, scientists studying dark technological and scientific frontiers in psychical and paranormal, and extradimensional topics.
You have good witches, good wizards, and good magi as well, but they are sprinkled around sparsely, outside the annals of worldly power, being kind to children and devoting themselves to their crafts, refusing to be corrupted.