Here is the non-canon information of the types of witches and wizards that are in the Little Witch Academia universe.
Chemists, or sometimes Alchemists, are magickers that participate in chemistry by making potions and medicine.
Sucy Manbaravan is the known Chemist witch in the Little Witch Academia universe, because of her frequent testing of many types of poisons and cures.
Criminals are magickers who break the laws of magic, and are claimed to be guilty. Those who practiced forbidden magic (example: necromancy, worshipping Satanic religion) are considered to be criminals of sorcery. This is unlike the natural crimes of stealing and destroying public property.
The Little Witch Academia series have Luna Nova Academy to forbid criminal magic. If a witch participated any of the criminal magic, she would be tortured, banished, or executed as punishment.
Fadans are natural-born magickers who have the supernatural ability to communicate with fairies, sprites, and other mystical creatures.
Lotte Jansson is known to be a Fadan witch, because of her communication with several fairies from items by chanting a signature hymn from her home country. She doesn't use her song all the time when she talks to the Will-o-Wisps and other fairies in free mind.
Faunistics are natural-born magickers who have the supernatural ability to communicate with animals, regular and enchanted. There are some witches who understand animal language, but they are not Faunistics.
Martials are sorcerers with the mixture of martial arts and magic. Many known martial witches are mostly from Middle Eastern countries, Italy, Brazil, and many Asian countries.
There are probably no known witches who are in this group in the original LWA series.
Paladins, frequently called paladin witches, paladin wizards, or paladin sorcerers, are the warrior-type magickers that are the sub-format of any type of sorcerer. The appearance of a paladin sorcerer seems to have a magical-girl/boy aspect. Each paladin sorcerer comes with one of the sixteen supernatural elements.
Psychonics are natural-born sorcerers that are given with one aspect to control. They do not need magical wands, unless they are tasked to dwell another. Some Psychonics have their eyes glowing into the element's particular color if they use their aspect for a long period of time. Any sorcerer who gain magical armor may not be a Psychonic, but they get to control their element in combat.
Some have alternate titles for a particular element other than just a psychonic: there are aquatics (water psychonics), psionics (lightning/beam psychonics), pyroes (fire psychonics), cryoes (ice psychonics), terranians (earth/nature/wood psychonics), and aerials (wind psychonics)
There aren't any known psychonics in the Little Witch Academia story.