Stop Safari from automatically logging me into Gmail

Hi,

I just got a new machine (OS X El Capitan v.10.11.5) at work a few days ago, and Safari (v.9.1.1.) keeps me logged into my Gmail account unless I explicitly log out of it before quitting the application. Other people rarely use my machine, but I still like to make sure that nobody can accidentally view my personal emails.

I did the following in hopes of stopping Safari from automatically logging me into the account:

- Safari Preferences > Passwords > unchecked "AutoFill user names and password"

- Safari Preferences > AutoFill > unchecked all of the boxes

- Password is never saved for account.google.com

- System Preferences > Users & Groups > made sure that nothing is listed under "Login Items"

- I cleared all history, cache, cookies for Safari.

The first time I launched the browser after I cleared history, etc, I was not logged in automatically. But when I closed the application without logging out of the account, the same problem persisted.

Are there any other places I need to change the settings? Any advise would be appreciated.

Thanks