Lots of confusion can exist when talking about this in such general terms. It depends on what you mean when you say "connected to the internet." If you were worried about something specific, then it would probably be easier for you to just explain what that is.
In an attempt to clear things up a bit: that network icon showing or not showing does not turn on/off your connection to the internet. That is only a graphical representation of your network status. The usual idea of broadband is "always on," which means it doesn't turn on and off. If you actually want to terminate your network connection when you log off, then you need to disable your network connection. This is not done automatically when you log out. This way, your PC is not sending or receiving packets while you are logged out. Make sense?