The easiest way to explain why some male deer are called stags, while some are called bucks is to refer to textbook definitions:

Stag is defined as the mature male deer of the genus cervus, such as cervus nippon (sika deer) and cervus elaphus (red deer). 

Buck is defined as the mature male deer of the genera capreolus (roe), dama (fallow), elaphodus (tufted-deer), hippocamelus (huemul), hydropotes (water-deer), mazama (brocket), muntiacus (muntjac), odecoileous (black-tailed, mule and white-tailed), ozotoceros (pampus), and pudu (pudu).

Both of these definitions are very formal, and are the ones I tend to use, because, of course, whenever you’re asked to explain something, you want to be able to have a clear definition to base your answer on.

Understandably, though, many people tend to use much looser definitions, such as saying that a stag is the adult male of any of the larger species of deer, which is consistent with saying that male red deer and sika deer are called stags.

But this ‘looser’ definition certainly does not work with fallow deer that is relatively large species, which are called bucks.

So, why is a comparatively large species of deer, such as the fallow, called a buck?

Going back in time... deer have an extensive history of being hunted by man. And in Britain, there was a time when only two species were considered to be worth the effort of the hunt - red deer and fallow deer.

And to distinguish between these two, different names were given to the male versions of these deer - stag for red, and buck for fallow.

These days it is more appropriate stick to the first definition, which is based on science, rather than old out-dated hunting terms.