in my universe, magicians other than witch(who uses different kind of magic called power, allowing them to peform everything-magic) are divided according to how they use their magic, not to the kind of spells they use. they all use fundamentally same source of magic, so it is not unusual for, as an example, poison magic to be used both by a wizard and a warlock-it would simply differ in how they are used. a wizard would study and remodel/invent poisons whereas a warlock would apply them to solving missions tasked by their clients.
the term 'job' in the definition of magician distinguishes characters who use magic to benefit from charaacters who are made of magic, like a witch's underling, or a cursed human, or natural whimsical people from other world and their descendents.
Krinket of Dream Realm is made by a witch named Yetbos and is 1% cotton and 99% magic. So she can use some magic as it is what she is made of, but she is not classified as magician because she does not use much of it for her job, which is normally counting sheeps, watching out for a cherub seeking human manipulation and keeping her boss in line. Well she could use magic to fulfil the objective of her job in emergency but since that's not the main part her job is non-magical.
Fotapun, aka Faceless, Voiceless Lady, is cursed to express emotions only through reflected faces. That is quite magical, but she does not want to do so, nor does she benefit from that. She would not be classified as a magician but rather a person who is suffering an unordinary disease which could be treated according to the kind.
H'tubo knows magic and use it according to her means. However, back in her home world, what she did was not particularlly seen as a professional job but rather a very ordinary thing that could be compared to writing a letter or vacuuming the house. So she may use complex magic of high level, but unless she use it thinking she's doing magic like people of this world do, she would not be classified as magician. People like her are often called terms such as 'pixie' or 'critter' in some regions, to distinguish from non-magical people. (Whether they like it definitely varies from individual to individual.)
Well...What is magic? That is a very hard question that competes with what is science. Some say they are recently discovered laws of universe, so that they are science. Some say they are laws of universe that are not science. Some say they are bug of universe so that fiddling around with them is taboo. But most people don't care about the definition. If something gives vibes of magic it is called magic. For reasons hard to actually explain with words, unlike early times of era of magic, people seem to view people who try to define magic tiring.
Not trying to distinguish magic from what is not magic sometimes causes silly phenomenons, such as people of mid-Dahlia accepting multiple sun rising as 'extraordinary weather'.