The dungeon terminology is kind of interesting when you think about it. The way we use it has become almost completely divorced from the origin of the word. I assume the "a labyrinthine subterranean setting" use comes from Dungeons and Dragons somehow. That's the only definition for this sense of the word I found, but even that's only a superficial description of the majority of what are called dungeons nowadays. I would say it's honestly an outdated and incorrect definition, because it doesn't contain anything inherent or important to a space getting described as a dungeon. There's a whole elaborate mythology to the concept that this doesn't touch on at all.
So without trying to go too deep into the history of the word and getting off track, what is a dungeon? It's not necessarily underground, but it is some sort of contained space. It's unusual for it to be outdoors, but far from unheard of. "Labyrinthine" is a good word, and I think gets to the heart of the traditional form and purpose of dungeons. The traditional purpose of an RPG dungeon is as an endurance challenge. The sprawling design is not necessarily to provide a challenge in itself, but to wear you down and whittle away at your resources as you attempt to find treasures and advance deeper into the dungeon. A dungeon crawl is about resource management. Resource management is not really a factor in Zelda after the first few games. It's most prominent in A Link to the Past, which actually uses MP in a very RPG-like fashion where you can run out and be unable to defeat the boss at the end of the dungeon. But most Zelda games don't have MP and even in the more difficult ones dwindling resources are not usually a big factor.
Series like Dragon Quest, Pokemon, and SMT sometimes include some light puzzle-solving elements, like a room where you need to turn some statues or raise and lower gates or push some things, or a maze of teleporters that you need to understand the workings of to get where you need to go, and it's my understanding that this is inherited from the concept of tabletop dungeons having Indiana Jones-like riddles in them. It's from things like this that you can sort of begin to see where Zelda dungeons come from, but it's also important to note that these are not the default states of these games. A Dragon Quest dungeon is most commonly a fairly straightforward delve with some branching paths for treasure and dead ends to further wear you down with random encounters. Simple and functional. SMT on the other hand likes its elaborate nightmare dungeons, but the "puzzles" are generally mostly about wearing you down in a more elaborate fashion, which is why a lot of them can only really be solved through trial and error. I think a common flaw with later RPG dungeons is that when they do try to include puzzles, they're mostly uninteresting busywork or annoyances where you have to play Sokoban or a Hamiltonian Path or some other stock puzzle you've seen a million times before. Zelda was able to gradually include more complicated and contemplative puzzle-solving in large part because it dropped that element of wearing you down over time completely.
It's worth noting that the "dungeon item" concept didn't exist until Link's Awakening, which is the point in the series where it completely drops endurance testing. Every dungeon in the early games had at least one item to find, but it didn't necessarily have any use in that dungeon, and was sometimes completely optional. You could say that The Legend of Zelda began as a conversion of the tabletop RPG that tried to do away with number abstractions as much as possible to make an action game instead. The way I like to think of it though is that, you have Adventure games, which descend from Colossal Cave Adventure. They're generally about picking up items and figuring out where to go use them. But then you have Action Adventure games, which is kind of a wastebasket taxon nowadays, but in my mind Zelda is the defining Action Adventure game. There's a lot of talk about Zelda's "lock and key design", and that's how Adventure games work on a basic level. The items could functionally all be reskinned keys. But what makes an Action Adventure game? The items are actions. You simply cannot finish Zelda without getting good at using the sword yourself. Many of its items require some level of mastery and punish inept usage as much as unwise usage. And as the series goes on, the functionality of its items becomes more and more involved, with each one being a whole new toy to play with and introducing a new mechanic to be the focus of that level and referred back to throughout the game thereafter.
What a Zelda dungeon is has changed over time. At first they were really just labyrinths full of monsters, with the only puzzles being like "the room is dark" or "find which block to push to open the door". A Link to the Past dungeons could now have multiple floors and rooms with layers and whatever shape, and perhaps as a result of this are each differentiated primarily by having a distinct structure. The Desert Palace has the huge open first section, and then the narrow linear second section. Skull Woods has a bunch of different entrances, you make circuits around the Ice Palace, Swamp Palace needs you to drain water from rooms to access the lower portions, etc. This is where they first really had unique theming, but it's very light and doesn't really apply to anything besides the tilesets yet. Link's Awakening dungeons solidified Metroidvania level design almost a full year before Super Metroid. Each dungeon has a second "act" where after you obtain the item, you can go back through previous rooms and interact with a bunch of objects or obstacles in a new way. For instance, being able to smash the crystals after getting the Pegasus Boots in Key Cavern.
Ocarina of Time is where theming of dungeons truly became important. Not only did they all take on elemental themes, but they each had unique music reflecting their separate identities. The switch to 3D brought a somewhat slower and less combat-focused pace, as well as a naturally expanded capacity for puzzle design where you could now look around and interact with the whole space of a room. Perhaps the biggest impact it had for the series going forward is introducing puzzle box dungeons like the Water Temple, which have you manipulating the space of the dungeon itself to progress. Link's Awakening sort of had a precursor to this in Eagle's Tower, but it was a one-time puzzle rather than a switch that must be pressed with intentionality to change the dungeon's state. It was another increase in dungeon complexity, and each one was made more individually memorable by giving it a stronger identity and having you spend more time in them.
Wind Waker was another major shift for how a Zelda dungeon would be designed, and it's where they reached what was more or less their final form. The N64 Zelda games put a lot of effort into theming, but this still didn't quite apply to the mechanical contents of the dungeon. A lot of the smaller puzzles especially were at the end of the day still mostly about generic switches and blocks in some fashion, and they were also a bit random. The Forest Temple for example is full of very disconnected sections and ideas like shooting the ghost paintings, the courtyard, the rotating room, the corkscrew hallways, the block maze, and the falling ceiling. Wind Waker began designing its puzzles like a Mario stage. A mechanic is introduced, iterated upon, and twisted as the dungeon goes on. So in Forbidden Woods, you immediately learn that these eyeball plants blocking the doors can be destroyed by throwing these big nuts at them. A bunch of puzzles are made about getting these two together. Then you get the boomerang, which can destroy these without all that complication where none of your other weapons could, giving you that classic feeling of power. In the Earth Temple, beams of light can destroy statues and freeze the undead, and your partner Medli carries a reflective object which can redirect them. Then you get the Mirror Shield, allowing you to not only more easily utilize light beams without having to pose Medli like a doll, but solve more complex light puzzles where Medli reflects the light into a specific spot for Link to stand and redirect it a second time somewhere else.
If there's one aspect of the series that even the best imitators usually fail to replicate, it's the dungeons. Almost never does somebody truly get why these are compelling. And to be fair, I think that's to be expected. Their nature has changed a lot between entries, it's hard to have a good picture of what they really mean without analyzing the whole series. The original game, A Link to the Past, and Majora's Mask have almost nothing in common in their dungeon design philosophy. But that fans are continually disappointed by outside attempts at crafting them points to there being some sort of unified expectations.
Dungeons are decently large, and usually non-linear (in the sense of not being a straight path that largely avoids backtracking to previous rooms), or at least attempting to give the illusion of non-linearity. Zelda will occasionally make purposeful use of a more straightforward level that is technically a dungeon, like Hyrule Castle in A Link the Past, but this is an exception, not the norm. Also present is the expectation that you have to think of the dungeon as more than the room you are currently in. In its most famous form, this concept was codified by Link's Awakening, which began the tradition that the item in each dungeon would be used to interact with things in previous rooms in new ways, and eventually defeat the boss. But even the original game's simple labyrinths have a shape which can serve as a clue to how to solve them, and ask you to figure out how to navigate from one part to another, because you can't always just walk straight there. A Link to the Past has you figuring out how to reach a particular room in a more involved version of the same process, now capable of using multiple layers, floors, and more unique layouts to complicate things.
This is the defining factor of the Zelda dungeon experience from The Legend of Zelda to A Link Between Worlds, and it's different from a standard RPG dungeon because there is no expectation there that actually finding the end is a challenge of anything other than persistance. Again, RPG dungeons are made first and foremost to wear you down. Most don't feel any need to make a puzzle of it and hit you with constant locked doors and secret passages and whatnot because it's plain overkill for accomplishing their purpose of making you walk for long enough to get a bunch of random encounters and make decisions about looking for treasure or rushing to the end, item use, when to heal, whether or not to leave and come back later, etc.
That basic pathfinding experience is at the heart of a Zelda dungeon. It's not just a series of puzzles and monster encounters with a map, a compass, an item, and a boss at the end. The puzzles and combat are, especially early in the series, extremely simple. They're merely the spice to keep you engaged in the moment to moment as you work on the larger navigational problem that is the true meat of the dungeon. With that perspective, people who try to imitate Zelda dungeons often come off like someone making a Mario course with enemies and blocks, but no platforming mechanic to serve as the central thrust of the level. In the 2000's, the series made a shift towards having a greater focus on individual puzzle motifs which grow more complex room by room, often at the expense of the original focus on navigation, but even this has a level of complexity that others usually fail to replicate in how it interacts with the theming and items.
Take Okami, for instance. It's very good at items and exploration, but in terms of dungeons, not a single one would ever make the cut in an actual Zelda game. Most pressingly, most are far too short. Some don't even have a boss. They're also mostly straight paths, and this is a little odd. The Moon Cave, the most time-sensitive dungeon plot-wise, is the only one to have a sprawling layout somewhat akin to a proper Zelda dungeon. The climactic Oni Island being a gauntlet of challenges sort of works, but when the rest of the game is just as focused, it doesn't feel like an intentional deviation in pacing and more like business as usual.
The reason people often talk about open air Zelda as if it doesn't even have dungeons is because in any meaningful sense, it doesn't. The Divine Beasts and the TotK Temples are slightly more elaborate shrines, but being ultimately just a few "rooms", are still far too small and simple to really replicate the sort of experience any past dungeons gave. Hyrule Castle is a large explorable space, but like most of the game, it puts no real obstacles in your path, so finding your way to the top is more of a formality than something meant as an actual challenge. I think they may have tried to do navigational challenges with things like the different entrances you need to find in the Wind Temple, but the "how do I get there" always ends up coming off as more of an isolated puzzle no different than "how do I get past those flames" or "how do I open that door" due to the small space.
