I am not familiar with the tutorial you are referring to, but "the Host player (i.e. developer of the game) maintains an-always running version of the game online as a host server" is correct, if you intend to have an always online dedicated host.
The signalling server is indeed an always running server, but it does not run your game. It's purpose is signalling, which is a method to connect players over the internet. It would indeed not solve your problem. The Construct team already hosts a signalling server for free which you can utilize, and is generally sufficient for most people's use.
A "server" can be any computer that is usually always on and connected to the internet. This can be your home computer, a computer in a data center, or a rented computer from a bigger company like Amazon or Microsoft. A server can run many things. In your case, you want a dedicated host version of your game running on a server, so people can connect to it.