methyl-cytosine

You mean methyl-cytosine; methyl-cysteine would be a derivative of the amino acid cysteine.

The authors find that methyl-adenine seems to play a role opposite to methyl-cytosine, surprisingly.

The DNA base cytosine is often modified to yield methyl-cytosine, a methylated base with a well established role in the regulation of gene expression; methyl-cytosine tends to mark genome areas with lower probabilities of gene expression.

I agree that it could be useful to introduce students to the concept of epigenetics, of which base pair modifications such as methyl-cytosine are just one aspect.

For example, in bacteria, A's are often methylated, and in Humans, C can have all kinds of different modifications (5-methyl-cytosine, 5-hydroxymethyl-cytosine, etc.

As Watson and Crick so famously understated in their seminal paper, 'It has not escaped our notice that the specific pairing we have postulated immediately suggests a possible copying mechanism for the genetic material.' When DNA is replicated, methyl-cytosine pairs with guanine just like cytosine does.

So your answer is either "because you're not reading the right books" or possibly "because methyl-cytosine is clearly a variety of cytosine, which is one of the four 'classic' bases".

"In the early 80s, to these four "classic" bases of DNA was added a fifth: the methyl-cytosine (mC) derived from cytosine." How come any biology course or book I've ever seen, whether high-school or top-university level, only teaches about four bases?

Methyl-cytosine is a modified version of cytosine that is a very important part of epigenetics, but since it is almost irrelevant to other aspects of genetics, I would argue that you could not consider it a 'fifth base'.