I dont think you using 2 total different accounts or there might be something wrong with your lan network.
Steam wouldnt kick you for 2 seperate accounts both logged in.
It will only kick you if you use the same account because of obvious reason.
The reason why the game itself closed is most likely because your steam account was logged off making steam close every app that was opened with steam.
My advice would be to check the accounts that log in on the desktop and laptop and compare them.
Another thing that could be it is a complete different thing but thats that both laptop and desktop used the same ip adress from your local network, kicking you off from your connection and gave the internet to the desktop.
Would say check your network configuration if you both use the same ip adress by accident.
If this doesnt solve the issue i dont know but its what i would do first.
Ps. When this happens do you get any pop ups with errors or does it display anything at all? This way we can get a better understanding of the issue.