Let me try to sum this up a little to help:
What that entire statement is referring to is peer to peer networking on a hub and spoke system.
All clients calculate there turns, moves, actions and reactions on there own machine. The server stores nothing, it just acts as a catalist to hold the clients together.
Instead of 1 client being the server and then having a list of clients to react and drop to in the event of a lost client / server scenario, the server itself is the glue that holds the set together.
It has no business / game logic, it doesn’t need it. All it needs is a way to know who has connected to it, as long as player count > 1 then game is still active. All players get the other data from the other players bridged through the server.
Let me analogize this.
Peer - Peer / pure: hot potatoe
Peer - Peer / hybrid: (3rd person): duck duck goose
Hot Potatoe:
Entire thing has one potatoe, it gets tossed around, one master controls the tossing of the potatoe, that master goes away, the potatoe hits the floor.
Duck Duck Goose
The controler is a 3rd party, the child walking around the circle per say. If a player drops out, the 3rd party is still there and skips that empty slot and duck duck on.
You start the 3rd party mode server, connect the clients, and still are working in a peer to peer situation. As long as someone is connected, the data about the entire game exists somewhere. You want to stop the game in the event player count <2. This mode is best suited for games like Red Alert, Grand Theft Auto, well, anything under 32 players. Thats probably pushing the barrel.
Lets look at Red Alert as a 3rd party mode server, you start the server, you connect 4 players to it, the server is a broadcast, thats all. Player 1 does his work on his pc, and all logic about his work stays on his pc, he broadcasts to the server what he alone has done, the server broadcasts to the other 3 players what he has done, there machines adjust his work accordingly.
The server has no care nor any logic if the work the players are doing is good or bad, it doesn’t even care if they cheat. Its just an anouncer, thats it.
Hope this helps.