You are misunderstanding the use of word server.
HostGator hosts websites. When a user goes to this website (with your game on it), the game gets downloaded into their local computer and run from there.
A multiplayer game server would be a dedicated computer that's always on and connected to the internet, preferablly on a fast/stable connection. You would use this computer to go to the website where you game is located and join a room first, as the host. Since your server is always connected as the host, everyone else who joins would be a peer.
There are many different kinds of servers. HostGator is a web server. A web server won't be able to run your game, it can only serve it to users. The threads you mention are describing a game servers. A game server can be any computer that has an operating system that can run a browser. Could be one in your house, or office, or one that you rent.