What's up with the "Stay signed in" box with the check mark? It doesn't work either way?????

If you Click on the check mark or Not you get the same results. If your browser comes down for some reason, even if you X'd out of it, I was under the impression that Either Clicking the box or Not (at least one of them) would give me the results I want, which is this....

If for some reason the browser came down, even if I accidentally X'd out of it, I DON'T Want my Yahoo E-mail account to automatically come up logged on if the Browser drops for any reason.

So... I thought maybe I misunderstood how it worked and Didn't click on the box at Yahoo Login and X'd out of it and the results were the same.

I almost Always Sign out when I'm done but I was in a hurry and just X'd out of the session (thinking No Problem, I clicked on the Stay signed in box but several hours later when I clicked on the "//http:mail.yahoo.com" URL my Yahoo E-mail account automatically came up.

What's Up with that?