For me, a witch dabbles a little in all those things, but doesn't specialize in any of them per se. A witch does read omens, but they also lay hexs, cast charms, brew potions, summon monsters, and turn people into toads. The schools of magic kinda assume you're going to focus on one of those aspects, rather than touch on several.
Furthermore, if a "witch" is just a wizard specializing in a normal school of magic, there is nothing "witchy" about their magic and you've gone to the Harry Potter school of "witch is the feminine of wizard" rather than make witch something unique.
What makes a spellcaster "witchy" is their motivation, methodology, and personal style. Not their base skillset. What differentiates an hunter or soldier from a villain in a slasher movie is their choice of prey...plus maybe some scarring, deformity, and history of inbreeding. You have all sorts of creative control of your character even as a player. For example:
Character A: Studied and in her off time still teaches at a well-known College of Magic. Absent-minded, always buried in books, shies away from physical confrontation. She likes big, flashy spells like fireball but tries to avoid the squickier forms of magic like necromancy. She's mostly concerned with learning new things, ostensibly for the benefit of society; and seeks out other (civilized) wizards or ancient libraries from which to learn new spells, outside of her experiments in clean, climate-controlled laboratories. Somebody roleplaying this character might describe her spellcasting with fumbling for scrolls (to read incantations from) and other components from enormous sheafs tucked into her backpack, not because this is mechanically necessary but because it's the way the
character learned how to practice magic; maybe she uses the classic wizard's wand as a focus.
Sound very witchy? How about this:
Character B: An embittered, crass, and cynical recluse who lives off in the woods alone. Once a famous beauty from a wealthy family, she squandered her youth; and has now come to hate the society that scorned her as youth faded and wealth was wasted (or maybe even stolen by a faithless suitor?). She now looks like an old and weathered hag, with stringy, balding hair and sagging skin. Having no formal education, she taught herself magic by seeking out supernatural creatures and bargaining with or threatening them for secrets; some were nature spirits, but some were undead and/or fiends as well; and that's still how she tries to gain knowledge outside of some experiments on captive animals and people that she believes no one will miss. As survival rates aren't too high, frequent replacements are needed. Her goals are to find some magical way to restore her beauty and youthful vigor; she's not too particular on what the cost might be to others. She favors magic that makes others suffer, helps her learn secrets that others don't want her to know, helps her change how things appear; but disdains evocations and flashier magic as brutish and insubtle (also: secretly because she feels unsatisfied when enemies just die). Somebody roleplaying this character might describe her spellcasting using bloodletting or sacrificing small animals and/or their organs ("Eye of newt, a dog's liver"); or sometimes drinking some foul-spelling brew or applying greasy ointment - not because this is mechanically necessary but because it's the way the
character learned how to practice magic; maybe she uses a spell component pouch filled with dried herbs and dripping animal organs as her arcane focus.
Even if you don't like the idea of "witch as merely the feminine version of wizard", there are undeniably characters within the Harry Potter-verse that fit the classic "witch" archetype. An once again, there's nothing whatsoever that keeps members of the wizard class from practicing different forms of magic. Their subclass specialization might merely represent inborn talent in a particular area or extra secrets uncovered while they were learning magic rather than the focus of their career. A specialist wizard might not even LIKE the school they are technically focused on, choosing very few of them.
My issue is that warlocks have too few spell slots (due to how they structured Pact Magic) and they end up relying on cantrips and invocations, and that creates the illusion of doing 1-2 things constantly and a few "whammo" effects ever-so-often. I think the class is fun for certain playstyles, but I think warlocks are really geared toward an "attacker-with-tricks" playstyle moreso than a dedicated spellcaster (so much so, I tend to thing of them more like the magic-equivalent to rogues than akin to wizards or sorcerers).
Yes, warlocks rely frequently on invocations and cantrips when their spell slots run out. However, one might spend most of their time using Misty Visions to confuse and misdirect enemies, and/or the Frostbite spell to inhibit counterattacks; another might focus on adding debuffs to Eldritch Blast; a third might primarily be interested in raw damage (which doesn't seem very "witchy" to
me personally).
I built a shadow sorcerer. My biggest problem is that anything I wanted to do with them, I tended to find there wasn't enough spells to support the concept. They are lousy necromancers, have no summonings, not a lot of illusions, and very few debuffs. Coupled with their low spell selection, I found that they ended up still heavy on blast magic with a few key utility spells scattered through. But that's a topic for another day.
Limited spell selection is a classic problem with or feature of the sorcerer class. Careful spell and metamagic choice is the answer to this; and frequently you'll have only ever have maybe one or two spells for an entire category of magic; which is a different mindset than many other casters. It's still quite easy to support a witch concept - I can provide an example if you're actually interested in one. It's true that that they can't do summoning - but that's not a critical part of the "witch" archetype IMO. They don't have some of the highest-end illusions like Programmed Image or Mirage Arcane, but they have plenty of low-level stuff that works just fine. Phantasmal Force is a very good candidate for twinning btw. They have enough debuffs to make the category functional. Keep in mind their limited spell selection has expanded power and utility due to metamagic. A low or mid-level debuff that inflicts disadvantage on the save via Heightened Spell is frequently more effective than a higher-level spell.
