Look at what you’ve written; this time, you’ve actually gotten good sources, including a precise definition of ex cathedra. What’s confusing?
From RCC web site:
Ex Cathedra
Literally “from the chair”, a theological term which signifies authoritative teaching and is more particularly applied to the definitions given by the Roman pontiff. Originally the name of the seat occupied by a professor or a bishop, cathedra was used later on to denote the magisterium, or teaching authority. The phrase ex cathedra occurs in the writings of the medieval theologians, and more frequently in the discussions which arose after the Reformation in regard to the papal prerogatives. But its present meaning was formally determined by the Vatican Council, Sess. IV, Const. de Ecclesiâ Christi, c. iv: “We teach and define that it is a dogma Divinely revealed that the Roman pontiff when he speaks ex cathedra, that is when in discharge of the office of pastor and doctor of all Christians, by virtue of his supreme Apostolic authority, he defines a doctrine regarding faith or morals to be held by the universal Church, by the Divine assistance promised to him in Blessed Peter, is possessed of that infallibility with which the Divine Redeemer willed that his Church should be endowed in defining doctrine regarding faith or morals, and that therefore such definitions of the Roman pontiff are of themselves and not from the consent of the Church irreformable.” (See INFALLIBILITY; POPE.)
It is well further to explain:
* that infallibility means more than exemption from actual error; it means exemption from the possibility of error;
* that it does not require holiness of life, much less imply impeccability in its organs; sinful and wicked men may be God’s agents in defining infallibly;
* and finally that the validity of the Divine guarantee is independent of the fallible arguments upon which a definitive decision may be based, and of the possibly unworthy human motives that in cases of strife may appear to have influenced the result. It is the definitive result itself, and it alone, that is guaranteed to be infallible, not the preliminary stages by which it is reached.
in Session IV, cap. 4, where it is defined that the Roman pontiff when he teaches ex cathedra “enjoys, by reason of the Divine assistance promised to him in blessed Peter, that infallibility with which the Divine Redeemer wished His Church to be endowed in defining doctrine regarding faith and morals”.
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