What is a wizard? You're probably thinking about this right now. When you overthink things like I do, meanings of concepts get overworked until they're not recognizable.
Nomenclature
In English (the official language of space and time according to Doctor Who), the "-ard" suffix is attached to the end of a word to denote when someone is of a condition. Example: dullard, drunkard, and wizard. If dullard comes from dull, and drunkard comes from drunk, where does wizard come from? Wiz?
Wise.
Literally, wizard means someone who is wise. A class of the wise. That's a heck of a label for a class. Is this the intelligentsia class?
Wizard as a word originally described a learned person. The association with magic or the occult came later, but that's outside my area of expertise.
Original D&D Classes - Origins of the Modern Wizard?
The original D&D game had only three classes: Fighting-Man which later became the fighter, Cleric, and Magic-User which has since become the wizard.
Classes in Advanced D&D 1e had level titles, which, at least aesthetically, resembled ranks. The titles got more prestigious as you leveled up. Fighter earned titles like soldier, knight, then lord (or lady). Clerics earned titles like acolyte, village priest, bishop, and patriarch. A magic-user enjoyed various titles that don't immediately denote more prestige, and they weren't called a wizard until level 11.
If the rules of the game describe the setting, and by some interpretations they do, classes might describe social classes within the setting rather than just character classifications within the rules. This is actually a very good way to ground the mechanics of your game within the world. Fighter resembles the military class. Clerics resemble the clergy class. Magic-Users resemble the, what, class of scholars? Arcane? Mystics? Sure.
In classic D&D, there was an assumption that when you reached your level cap, the game changed from adventuring to domain play. I have no idea how it worked, but each class secures land, builds its own stronghold, and earns its own followers.
Here's the caveat, there appears to be differences in roles for domain play depending on your class.
While fighters were the sort of military commander, the wizard was more of an advisor or strategist. The sounds like a mentor archetype in fiction. The mentor character is typically a supporting character in fiction; not the main character. Oof. Ouch, Gary.
What is a Wizard, Really?
See, here's the thing. There's no real-world equivalent for a magic-user or fantasy wizard. It's made up. That makes it an abstract concept. Even if we all think we understand what a wizard is, someone else can come along and redefine what a wizard is for their game or fiction, completely turning the concept on its head and subverting all conventions, and they're technically not wrong to do it. You would simply say their version of a wizard is different.
In fiction, wizards are typically the mentor, advisor, scholar, or villain rather than hero.
Who are the famous wizard-mentors of fantasy? Merlin. Gandalf. Dumbledore. Obi-Wan Kenobi. Yes, Obi-Wan counts! Watch Episode IV, A New Hope! He totally counts!
Also, Dumbledore is in there now. Good on you, JK Rowling. You know you are successful as a writer when one of your characters can be counted among classical archetypes. Respect.
What happens when the full-fledged wizard becomes the protagonist?
Doctor Who?
The Doctor is a fully-fledged wizard-protagonist. He's a scholar, a traveler, a keeper of obscure knowledge, someone who understands things ordinary people do not, he solves problems through knowledge rather than force, he mentors younger companions, and he seems mysterious and supernatural from the perspective of ordinary people.
Just replace his sonic screwdriver with a magic wand. Replace the TARDIS with a wizard's tower. He doesn't perform spells, but he has the wisdom to perform scientific feats that are like magic to others because no one else can replicate them.
D&D says wizards are magic artillery. Literature says wizards are wise.
The Doctor is often the most powerful person in the room because he understands the situation better than anyone else. That is a wizard according to the classical definition.
Magic isn't what makes a wizard. Wisdom is. It's in the name!
What Kind of Magic-User Are You?
Circling back to classic D&D, the old name for the wizard class was Magic-User and wizard was a high-level class title. They chose magic-user as a class name because they weren't concerned with what kind of magic-user you were, like illusionist, necromancer, wizard, warlock, or sorcerer. They were only concerned that you were a magic-user.
In this line of reasoning, what kind of wisdom does your wizard possess? What kind of wizard are you, and how does your wisdom fit into the setting? Are you an occult scholar? A sage? A scientist? A strategist? A seer? A teacher? A healer? Yes, I'm counting healers!
D&D 5e Wizards Are Still Wizards (Kind of)
Modern D&D distinguishes wizards from other spellcasting classes by the source of their magic. Wizards learn spells through study, practice, and mastery. They can research spells and add them to their spellbooks. They're categorized into arcane traditions which focus on specific schools of magic (abjuration, conjuration, divination, enchantment, evocation, illusion, necromancy, and transmutation).
The D&D 5e wizard is relatively in line with the concept of a wiz-ard. I would tweak it by giving them more utility spells and less damaging spells.
Conclusion
The identity of the wizard is wise one rather than powerful one. You're more than your spells. When you play a wizard, figure out how you fit into the setting. You start out as an apprentice wizard at low levels. Your experience makes you wiser. Your growing power is an expression of that wizening. By higher levels, get an NPC apprentice. Pass it down. Your wisdom is who you are. It's your legacy. And put a little bit of the Doctor in your character!