Quite simply, ex cathedra describes the Infallible Extraordinary Magisterium.

Well, in more recent times we have a very clear rule of thumb.  If something appears in the Acta Apostolicae Sedis, it's the intention of the Pope to act as Pope.  These aren't the Acta Jorge Bergoliensis, but Acts of the See itself.  Essentially it depends on the intention of the Pope to be acting in his capacity as Pope, vs., in this case, as Jorge Bergoglio.  In the past, there was some debate about inferring that intention, often due to the nature of a docuмent (Bull, Encyclical, etc.), but with the creation of AAS, any ambiguity is cleared up.

Simply because the Pope is teaching non-infallibly, say, in an Encyclical letter, does not translate to him acting as a "Private Person".  He's clearly acting in his official capacity as Pope.

That's actually all that the term ex cathedra actually means.  This term has been turned into a shorthand for when the Pope is acting infallibly, but that's not technically correct.  Acting ex cathedra is simply ONE of the notes of infallibility.

Here's from Pastor Aeternus:
That parenthetical "that is" actually defines the term Ex Cathedra, and private person is the opposite of Ex Cathedra, namely when he's NOT exercising his office as shepherd and teacher of all nations.  So even in a simple non-infallible Encyclical letter, he's still exercising his office as shepherd and teacher of all Christians, but not necessarily defining a doctrine of faith and morals ... etc. etc.

In the past, things where the Pope might be writing a letter to an individual bishop, or giving a sermon, or opining privately (such as when Jorge is answering questions on his plane), those are acts generally construed as the Pope acting as a private theologian, since they're not addressed authoritatively to all Christians.  But, as I said, in recent days, the inclusion of something in AAS has served to remove all ambiguity and guessing regarding his intentions.