A Gamemaster by Any Other Name...
When it comes to roles at the table, the player is clearly defined. A player's task is to portray their character and engage with the world and situations created by the game master. However, the game master's role is much less clearly defined and much more variable depending on the game being run. In short, the game master is a worldbuilder, actor, adjudicator, narrator, facilitator, and storyteller, with varying levels of each depending on the person and style of the game. While game master is an adequate title to describe all of these roles in a complete package, it obfuscates each of the parts that make up the whole. A game master running a dungeon crawl in D&D 1e will be focused on different roles than a game master running a political intrigue game in Vampire: The Masquerade, despite their titles being identical. In addition, to someone new to the world of TTRPGs, the title of game master gives little insight into their role without previous knowledge of how tabletop games work. The obvious solution to these issues is for systems to create terms that more specifically describe the role that the game master plays.
However, systems can go overboard with this idea, not solving any issues that the title of game master presents. The specifics of the role the Damn Pinkerton (Hawgbilly) or The Man (Starchildren) play are less clear than the simple title of game master. Without any knowledge of the game, one may assume that the Damn Pinkerton or The Man are character classes or villains to be faced. Despite these issues, system-specific titles set a tone and provide immersion into the world, something that "game master" doesn't do. However, if emersion is not a primary goal of the game, a middle ground can be struck that is more evocative than "game master" while expressing the specific roles that the game requires. I have identified six such titles, organized into 3 categories. This is in no way an exhaustive list, but an exploration into how different terms can change the expectations and flavor of a game.
Referee/Judge
These titles originally came from war games, in which the referee acts as a mediator between players and a source of both rules and rulings. When it comes to TTRPGs, the titles most often appear in old school games or their imitators, as their culture of play was directly inspired by the war game framework. This connection, however, is not the main reason that the title of referee is fitting for many games. Instead, the titles bring to focus the role of the adjudicator, interpreting the rules provided and making fair rulings from them. Any rules-heavy system could reasonably use referee or judge as their title for game master, although the connotations associated with these titles may not make them desirable. The titles imply an adversarial relationship between the game master and players. The main job of a referee in sports is to adjudicate rules by calling fouls and subsequently punish teams(players), while the main job of a judge is to hear a case between two lawyers (players) and determine which is correct. It follows that many early TTPRGs that adopted these terms also adopted a culture of conflict between players and the game master. Because of this unavoidable connotation, it is best that these titles be relegated to classic games while more inviting titles foster a better relationship between everyone at the table.
Host/Facilitator
Like judge and referee, these titles impose their neutrality. However, these titles achieve this by emphasizing the player's importance at the table. It is the role of the host to make players feel comfortable and able to express themselves freely. Because of this, the title lends itself to horror games and games with dark subject matters. When exploring dark themes, it is important that the game master does not needlessly make any players uncomfortable or afraid. The title of host expresses this caring role, while facilitator goes one step further, implying a game that is tailor-made to facilitate the needs and limitations of the players at the table. However, these titles also imply a passiveness in their roles, that the game master is only there to mediate a game between players, when in reality the game master plays the most active role at the table. In games where much of the plot is dictated by the players, this may not be a significant issue. For story-based games, games with rich worlds and plots, this incongruity means that these titles aren't apt.
Storyteller/Narrator
These titles emphasize the narrative roles of a game master: worldbuilding, roleplaying, describing, and plotting. While the titles of storyteller and narrator are similar, there is a key distinction that sets the terms apart. Narrators react to an unfolding story and describe it appropriately while storytellers actively forge the story, furthering a narrative plot throughout the game. This distinction means that "narrator" is fitting for sandbox-type games with a large amount of player choice, and "storyteller" is best suited for games focused around a single plotline, such as a mystery or heist. Regardless, both terms express a player-centric attitude, as there is no story to tell and nothing to narrate without players and their actions. However, they also imply a neutrality that is hard to achieve. The game master should be on the player's side, allowing for leniency and exceptions to the rule, but an objective narrator is on nobody's side. This can get in the way of the fun of the game by limiting the actions players can take based on how realistic they are rather than how interesting or impactful they are. In addition, the title of storyteller puts a large amount of onus on the game master to carefully craft a story for the players, when in reality, the story is created during the game through the collaboration of the game master and players.
Where does that leave us?
While none of these titles perfectly encapsulate all of the roles a game master performs at the table, each one provides insight into the priorities of the game and what the game master should focus their energy on. For my money, this imperfect description is much more useful than the perfect but generic title of game master. In fact, I would argue that game systems should retire the use of "game master" to refer to the people running the game. The title is a useful tool to neutrally discuss the practice of running games, but the descriptive titles above, or even tailor-made titles, clue the game master into the type of game they are running and, hence, how they should adapt their roles in order to match the system.