Different power options.

• Logging off closes all the applications being used and then go back to the log in screen that you see when you perform a startup
• Locking keeps everything open and requires that current users password be entered to regain use of the computer.
• Sleep is a low power state where that everything currently open is saved to disk and the screen goes blank. This uses very little power and when you wake the computer by mouse or keyboard action it is very fast to regain use of the computer.

<<"I understand the "Shut Down" button. ">>

I am not sure you do. There is no generally available single button that accomplishes an organized shut down for most Macs. In most cases, pressing and holding the power button causes an emergency power-off, not recommended for daily use because it does not "put away" the disk Drive directory properly.

Only choosing Shut Down off the  Menu causes a proper, organized shut down, and there is one-key keyboard shortcut.