Hi,
The batteries you buy over the counter or take out and recharge are the ones that should be stored apart from the camera.
Inside any computerised device, which has a clock running, there will be a small rechargeable battery that the device charges and that keeps the clock and calendar functions going. They have to run all the time for obvious reasons if you think about it; mostly because they change second by second...
But, the hard wired internal ones don't last for ever and might just need replacing eventually. The Nikon F6's one could be 15 years old and so has died*. It happens with cars, clocks, watches and so on. Most of them can only be charged a certain number of times and then fail and as they age they start to hold only a fraction of their original capacity.
As for lithium batteries, a lot of batteries come in lithium only and I've looked at old cameras that use them and found corroded battery terminals. But that could just be due to cheap and nasty non-lithium fakes...
Regards, David
* I don't know if the F6 has an internal one or a replaceable one. In rough terms digital cameras have internal ons and film ones have replaceable ones, like coins in size and shape. For example the Konica A4 has a CR123A powering it but the clock etc have a CR2025 powering the clock etc. The CR2025 fits in a small compartment in the camera back.