Prahlad provides an excellent answer on the origin of the term, but doesn't expand that much on the legal implications of using the term. In summary, it forces any modifications to copyleft works to also be released under the same terms.
First off, copyleft has no legal significance.
This means that if you were to say that a project is copyleft, it doesn't mean anything. Under the law, that statement isn't clear, and can't have any meaning, since copyleft is never defined under the law. Think of it as a term in popular culture: people use it, but it doesn't really mean much.
Second, there is no "one-size-fits-all" kind of copyleft.
Let's look at a few different "copyleft" licences: most notably, the Mozilla Public Licence, and the GNU General Public Licence. These are both "copyleft" licences - what's the difference?
A simple definition of copyleft is that changes will be released under the same licence. With the Mozilla licence, you can make your own projects under the Mozilla-licenced code, and your own code can be licenced under whatever.
With the GPL however, even if you make a function call to something under the GPL, your code has to be under the GPL too. If you do something like this:
Your code has to be under the GPL, because yo've now created a link to that code.
Copyleft and Share-Alike are basically synonyms.
The tag wiki excerpt, as you noted, says that the two terms are basically synonyms. We used to have different tags for both concepts, but realized that the question that the tags were applied to utilized both tags, and tried to convey the same meaning. There is a good key difference to note:
Copyleft is used to describe source code.
Share-Alike is used to describe works of art.
In fact, copyleft was used as a term originating with the Free Software Foundation, whereas share-alike was used in the Creative Commons family of licences.
Fundamentally, they are the same, but the two terms are applied to different works.
Why describe it as Copyleft if you need to add a licence to it anyway? Would you not be better off describing it as copyright with already-agreed-upon licences?
Now that we've determined that there are different kinds of copyleft, and that copyleft is not a legal term, we can now answer the overall question.
Licences often provide a series of different terms, generally providing rights that govern patents, to attribution, to disclaimers of warranties and liabilities. Copyleft is just a 'popular culture' term: it's just used to simplify most of this. But it can't be used as a substitute for a licence.
Concisely, we just use copyleft to provide a general summary of a key point of the licensing terms, those affecting distribution. Aside from that, it's up to the legalese in the licences.