What Is the Person Who Hosts a Game Called?

The title depends entirely on context:

General

• Host – Universal.
• Moderator – Facilitates, enforces rules (forums, live streams, tabletop).
• Facilitator – Workshops, team building, educational games.

Tabletop RPGs

• Game Master (GM) – Universal.
• Dungeon Master (DM) – Dungeons & Dragons specific.
• Storyteller – World of Darkness.
• Keeper – Call of Cthulhu.
• Referee – Old-school / OSR.

Game Shows / TV

• Host / Presenter – On-camera face.
• MC (Master of Ceremonies) – Events, tournaments.
• Announcer – Voice-over, prize reveals.
• Showrunner – Executive producer (behind scenes).

Card / Casino

• Dealer – Manages cards, chips, payouts.
• Croupier – European term (roulette).
• Pit Boss – Supervises table games.

Digital / Online

• Server Host / Admin – Dedicated server / listen server owner.
• Lobby Leader – Party creator in matchmaking.
• Streamer / Content Creator – Hosts community games.

Use the term matching the setting's tradition.