Magic Users are individuals who possess the ability to use magic.
Wizards are seen more as a race or species than a group. Wizards are exclusively male, and were born with their abilities via genetics. A natural magic ability is called Witchcraft.
Witches are similar to Wizards in having genetically determined magical ability, the only difference being they are exclusively female.
Warlocks are a unisex group of magic users that have been given their abilities later in life either by a powerful magic user, a spell, or some other force that turned them from a non magical user to a magical user. A Sorcerer can become a Warlock
Sorcerers are knowledge based magic users. They do not naturally have the ability to cast magic and instead study the arts of summoning magic from the universe, this is called sorcery. A Sorcerer cannot use magic non verbally and must use incantations or gestures to perform magic. They also are required to use wands, staffs, or other magical items to perform magic. Only the most experienced in sorcery can use magic with only their hands and even then the magic is extremely limited. While other magic users can still practice sorcery, it is very rare.
A mage is a magic user who believes in a higher power that gifts them their abilities. A mage is seen as more of a religion than a species or group. Mages follow a specific code and have their own places of worship. Mages often use their powers to heal or benefit the world around them as part of their beliefs. While mages are usually Wizards, Witches, and Warlocks, Sorcerers are rarely part of the Magi (the term for many Mages) as well
Magician is often an interchangeable term for a magic user, but Magicians generally do not have actual magic abilities and instead are avid enthusiasts of magic and may try and replicate magic through illusion or performance such as stage magician. Therefore, while Magicians are technically not Magic Users, much confusion can occur in the difference between stage magic and Magician as an interchangeable term for users of supernatural magic.