What you are doing IS being on the network, just not a huge corporate type network. A network is any set of devices connected together.
But your way is a bit roundabout, and a bit confusing with how you describe it.
Modem from the ISP goes to the Router, Router port 1 goes to the desktop, Router port 2 goes to the laptop, both laptop and desktop talk to the router to get online.
Or... do you just have a Modem that goes to your desktop and you are calling the modem a "Router"?