Login screen reappears after brief user inactivity

One of my older Windows 7 machines has two users: Administrator and Me. For as long as I've had it, Me is taken back to the login screen after a very short period of inactivity, perhaps 5 or 10 minutes. I cannot find anywhere to tell it to stop doing this.

There is no password set for Me, nor for Administrator.

When I say it takes me back to the login screen, I mean the icon for Me is displayed (only the icon for Me -- as opposed to the boot-up screen that shows the icons for both Administrator and Me) and below it is a button that says Switch User.

I just have to click the icon for Me to go back to my desktop or work, but it's a hassle.

For reasons I cannot understand, my two new Windows 7 machines are not doing that even though they, too, have two users, Administrator and Me, and those also do not have passwords. Very perplexing. I've looked in every place I can think of, from User Account settings to Power Options settings (which are now all set to "Samsung" profile for the SSDs I've installed in all these machines -- though I've then drilled through all conceivable settings within that and nothing seems to apply to this issue).

Thanks!

EDIT: I created a password for Me and that didn't help.