My ideal style of DMing is probably "contemporary." It's the style that goes something like:
* The players have limited control over the world. The DM sets the main elements in place, then allows the players to flesh them out as appropriate.
DM: Sir Huxley opens the chest. Inside, he sees a glittering weapon! Sir Huxley, describe the weapon to us.
Player: It's a battleaxe, with a double-edged blade.
DM: (writes "battleaxe +1" down on a slip of paper and passes it to the player)
* The DM uses theater of the mind to describe dungeons, corridors, rooms, and other scenery, and only resorts to using battle-mats, minis, and 3D terrain when absolutely necessary (cinematic battle scenes that are expected to last for more than an hour, for example). You don't need to draw and build out every single tunnel and door if they're only going to be background scenery.
* The DM talks to the players regularly between games and gets feedback about their characters, the game world, and their goals. Is a player satisfied with their character? Would it be okay to make a few changes? The next chapter of the story is going to focus on the warlock for a little while, is it okay if I bring in some of your bard's backstory as well? Are there any topics or subject matters that are off-limits for you? That sort of thing.
* The DM relies less on math and more on imagination when it comes to encounter balance. The DM trusts that the players will have the sense to flee from hopeless battles, and the players trust the DM to make escape possible (if also difficult and expensive).
* The DM isn't afraid to change or scuttle an entire adventure ("kill your darlings") and adapt the story if the dice or the players aren't cooperating. Sometimes the players don't want to go into the haunted house, and you're not going to make anyone happy by forcing them to do so. Move on!
* The DM doesn't just arbitrate the rules, but pays attention to the players as well. A good DM can "read the room" and notice if someone at the table is getting bored, angry, or uncomfortable, and will make course-corrections on the fly to ensure that everyone is having a good time. A good DM will take player complaints seriously and make adjustments.
Not sure if "contemporary" is the right word, now that I think about it. I mean, Dungeon Masters have probably been doing this for years, decades even, but this style of play didn't really become popular until the advent of live-play shows on the Internet became a thing. Maybe "modern" is a better word?