I have a button to the left of the Lock button, which "saves my session and puts my computer in a low power state so that I can quickly resume working." Hibernate, in other words.
No, hibernate is a state which requires no power. That's more like a glorified Sleep state.
Hibernate saves settings and open apps etc. to memory then shuts down the PC, and reloads them back into the OS upon boot. Saves re-opening stuff...