How does cold make batteries drain faster yet simultaneously they should be refrigerated to keep a charge? I have always known via common knowledge that batteries drain quicker when it's cold. I've been told it's best to keep them in an inner pocket close to body heat when outdoors (e.g., batteries for headlamps or cameras when one is hiking in the cold night). And I've witnessed this behavior myself as well as it did seem like my batteries drained quicker. Adding on to this, I've also heard that it can help to "prewarm" batteries.
Yet the other day, I also learned that a good method to preserve charge on batteries is to refrigerate them!
These two statements seem directly contradictory? Can someone explain to me how they're both true? For example: is it that when not being used in a device, a charge is best retained in cold conditions, but once in a device, cold makes batteries drain quicker (so the difference is being in use/in a device)? (With the add on that prewarming helps, hence the common advice to keep it close to body heat, since it's presuming that these will be batteries soon to be used?)
Or is it possibly that the two statements are for different types of batteries?
 A: 
I have always known via common knowledge that batteries drain quicker
when it's cold...

Yes, because lowering the ambient temperature causes chemical reactions to proceed more slowly, so a battery used at a low temperature produces less current than at a higher temperature. As cold batteries run down they quickly reach the point where they cannot deliver enough current to keep up with the demand.

Yet the other day, I also learned that a good method
to preserve charge on batteries is to refrigerate them! ...These two
statements seem directly contradictory? Can someone explain to me how
they're both true?

Yes, because refrigerating them slows down unproductive chemical reactions when the battery is not in use. But the  statements aren’t contradictory because even when when the battery is not in use, chemical reactions can still occur using up chemical energy. By refrigerating the batteries the consumption of chemical energy while the battery is not being used slows down.
Hope this helps.
