Half-Life doesn't work like some other games you might have played.
When you make a server, the people who connect are connecting directly to your computer, that's why if you have a slow connection, everyone will have high ping, etc.
There are only two types of servers, listen and dedicated servers.
Listen servers are made when you just create a normal game, and you play in it with other people who join.
Dedicated servers are when you check the "create dedicated server" checkbox before you make the game, or you configure and use a rented/owned computer meant to be used solely as a server. A "dedicated" server.
The thing about dedicated servers is you cannot play the game on the computer that is running the server, but this allows for less ping to all players, and better all-around playability (less chances of crashing, etc).