You can think of the MyBook essentially as a (very) big thumbdrive/USB key (or whatever you like to call them) - as Tony says it connects via USB to a single computer (although that computercan then share it out to the rest of your internal network as an SMB share), so if the machine is off then there will be no access to the drive. In some cases where your router/switch has a USB port, you can connect it there to act a bit more like a NAS, but that is a special case where the router is doing the sharing instead.
The MyCloud (or MyCloud Mirror, as you’ve posted it in this section) connects via ethernet, usually to the router or switch on the network rather than to a specific computer. It will then be visible and available to all of the computers on the network, to be used by any of them depending on how you’ve set up access permissions etc.
The MCM at least cannot be connected by USB to a computer. If you want to use it with a single machine, you have to connect it directly to the computers ethernet port via a LAN cable where it will appear on the machines network as a second machine in the same way as normal (but of course by doing it this way you may well not have access to anything else on your network as you may no longer be connected to it.
But if your internet is down, then you should still have full access to your local network via your router, which would include any network attached storage (NAS) such as a MCM - it’s essentially local cloud storage and has nothing to do with the internet at a basic level.