When I start the server on my PC, I can only connect to it by using the client.connect command. The server is not displayed in the game client. Is it possible to make the server visible in the server list without giving full access to its IP address? So that only I can see it and only I can connect too.

No, that's not possible. If the server is on the in-game list then it's public and the IP is visible. What you can do is add your local IP:queryport to your steam server favourites:

Steam > View > Game Servers > Favorites > +

Then you can connect directly from steam.