There could be a battery that this stuff is saved to, but looking at a diagram of the chassis in the service manual I'm not seeing one. There is a chip marked memory, so that's probably a RAM chip and it probably saves this stuff there. That chip may or may not need battery power to hold your settings when the AC is unplugged. It's worth checking for watch batteries inside our TV's and replacing them, but that's probably not the cause of the problem here. I say that because it is remembering what input you were on, so it's not losing everything when it's unplugged.
While looking through the service manual, I found that it does instruct to press mute then enter to save settings as Shawn suggests. Specifically, you do this while still in the service menu, then press power to exit the service menu.
I should have told you this before when I told you about the service menu, as I've done it before. I guess I forget about it, as I haven't used the service menu for anything except minor adjustments in about a year.
So, it seems like the TV will remember whatever settings you choose after you turn it off, but it only remembers them after unplugging if you "save to memory." Did you have your service menu settings written down? If not, I suggest you get them set up and then save them somewhere in case this happens again. Remember, you should write down all the default values as well, or you risk bricking the TV.