Both. Why not save power by turning off the router while shrinking the amount of time there is to break into my Wi-Fi network? I can't see a reason to leave my network on while I'm out of my house. I don't need it and as far as I can tell, my neighbors don't either.

That said if someone needed to use Wi-Fi, I'd be happy to open a guest network for them (I think my router can do that but I haven't checked).

>specific MAC addresses and make the access point invisible.

Both of these measures do nothing if any other device is currently connected to the wireless network. A passive attacker will still be able to see your access point (by inspecting packets sent over the air by other devices to the access point) and can spoof a MAC address to connect to it.