Regardless of if there were any windows open. I QUIT the app using Cmd+Q and the dot disappears below the Dock icon.
Now, I tried something yesterday, which was to remove from the dock and then reboot the computer. The app wasn't in the dock and it didn't auto-start.
However, this morning, something weird happened. I booted up again and the Safari app I had placed in the Dock again, but there was no white dot below. I popped out the room for 15 minutes and when I came back the white dot was once again there.
Not sure what could cause it to open in that time on it's own?
Rob