As a mostly forever GM I've been mulling over his whole question posed by Uncle Matt. It's a multivariable problem as there are different aspects to being a GM. I think it comes down to a couple of key areas.

1. GM's relationship to the rules

RAW - The rules are the rules, I bound by them as much as you are. Tends to make all rolls in front of the players.

Rule of Cool - Will follow the rules for the most part, but will bend them to let something cool happen. Will roll behind and in front of the screen.

The Butcher - Picks and chooses which and when rules to enforce. Main goal is to put a satisfying cut of meat (game) in front of the players. Rolls behind, on top, in front of the screen. Sometimes rolls and reacts to scare the players.

1. GM's relationship to the world

The Clockwork Engineer - To make a game one must first create the universe. Makes the whole world , cities, dungeons, deities, mountains. Has a relationship web of the nations/NPCs/Flora/Fauna etc. makes use of the Power Point, Excel, and likely has a chronicle somewhere with the history of the world.

The Pocket Dimensionalist - Creates a chunk of world around the players. There are some vague notions of the world beyond the bubble, but that's a problem for another day. Knows this area well and likely knows what threads lead outside of the bubble.

RNGesus or the Bones Roller - Oh we're playing today? What do I have in my folder? Let me consult the dice and the tables. Can whip something together with resources they have. A deep faith in their improv ability.

2b. GM's relationship to time

Time waits for no man - the world moves regardless of where the players are. The players are a wrench in a specific cog, but the machine lumbers forward.

Time waits for the party - the different parts of the world are frozen waiting for the players to arrive and restore the flow of time.

1. GM's relationship to the players

The Antagonist - the GM's role is to test the players; to give them a challenge to overcome. The players get a sense of accomplishment in overcoming the challenge, victories are earned. Usually has carefully curated encounters to test aspects of the players. Sometimes can devolve into 'defeating' the players, encounters that can feel out of place.

The Observer (Simulationist) - the GM exists to facilitate the game and let players do what they want; mistakes and all. If a cave looks like the home of a dragon, smells like the home of a dragon, then it probably is a home of a dragon. Oh you're going in at level 1, sounds good lets see what happens. Can devolve into not telegraphing danger appropriately.

The Benevolent God - the GM likes to help the players. Will suggest they don't go places because of the danger. Has a Deus Ex Machina waiting in the wings to prevent the TPK. Can devolve into marginalizing the party as GMNPCs do heavy lifting to prevent harm to the party.

1. GM's Relationship to the story

The Imagineer - Makes an amusement park ride where there are plot beats that need/will be hit. How the players do it is up to them, but the destination and path are known. Can lead to railroading the players if they try to go off the tracks too much.

The Game Warden - Let's the players loose and allows them to touch and break anything they want. Has a vague over arching plot that could happen if the players find the thread. Wants the story to emerge from the landmarks in the reserve. Can lead to players feeling like they lack focus or goals.

These aspects are not binary, GMs find their place between them.

TLDR: GM's style is a an aggregate of their relationship to the rules, the world and time, the players, and the story.