So, in general, there are like three ways to exit the game, and they affect how the green dots are set.
1) You log off. The green dot winks out pretty much immediately. (Note: The reason you never see this on Nutaku, is that Nutaku does not have a log-off option on the crossed-wrenches menu.)
2) You shutdown your browser without logging out. The green dot stays lit. You are still "logged on" until such time as the server decides to check up on you, for whatever arcane reason the server decides. When the server fails to get a response from you, it seems to log you out at that time, and your green dot goes out. This is very erratic. I have been "logged on" continuously for very long times this way, to the distress of players who think I am ignoring them.
3) You are away from your keyboard for an extended period of time. Your browser is still logged in. The green light will stay lit until something happens to the browser session. When your browser session drops, I believe you revert to option 2. I would often leave my browser logged in because I really liked our community, and I got to catch up on what everyone was saying if my session lasted until the morning. My players got used to not knowing if I was logged in or not, and were often surprised when I would suddenly jump into conversations when they thought I was sleeping. XD
I have retired from the game, but not from life.
In love with my Baby.