Actually the red dot means that you are online, but hiding that fact becuase you have chosen to be invisible.
Everyone else sees a grey dot under your name, only you see the red dot. I assume that administrators also see the red dot (what exactly would be the point of having code to make a red dot only to tell you that you are online - you knew that already!), but moderators do not.
If you were to log off and view one of your posts as a visitor, you would also see a grey dot.