Thank you both for the clarification!! When I came in now, I noticed that the "bottom" one wasn't checked. At first I was surprised and then I figured - oh, I bet MB saw my post and fixed it!
The fact that I came back that time and saw my name already listed, I assumed that by not logging out, I was listed as being there all the time. I didn't like that idea, so that's why I've made a point of unchecking it.
Does that mean, that if I do check it, that any time I come back, I'll be automatically recognized, time settings and new posts, etc. without having to log in? That WOULD be nice!