Preserve the desktop where an application's window was placed, after a system restart
Ok, I found a solution that's working for me, and now my application windows doesn't switch Desktop randomly on restart.

For the two desktop setup as in the question, first I go to the desktop where I want to stick an application window. Say, we want to stick web browser to desktop 1, we 


go to desktop 1, 
open the application on this desktop, 
right click on the application icon on the dock, then Options > Assign to > This desktop
similarly, stick applications on desktop 2


Now, after system restart, the applications are starting where they were sticked!