Here is a third explanation. Forgive me if I'm not a 100% correct. But, you can use your router. Even if it isn't connected to the internet. It works sorta like this.
Computer -> Router(Network) -> Internet
Step one you connect to your Router(Network). (If you have wireless desktop connection it'll appear just like wifi normally will, it just won't have internet) At this point you don't need to go further. Being on a network is like being connected via cable, except its wireless. So for example, sometimes my internet cuts out, but I can still connect to my dedicated server on my other laptop because its just the internet that went out. Not my network.
Computer 1 -V
Network -> Internet
Computer 2 -^
Dunno if that one explains it better.
I just feel it might need a better explanation. Granted this is just simply from what I understand, but it is how its worked for me the past five years.
Sorry if I'm wrong.