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.
So then, should anyone, which God forbid, have the temerity to reject this definition of ours: let him be anathema. (see Denziger §1839).
Vatican Council, Sess. IV , Const. de Ecclesiâ Christi, Chapter iv
This seems to fit St Peter's declaration, "You are the Christ, the Son of Living God", at Caeserea Philippi which was said "not from the consent of the Church", the other Bishops (or Apostles) put forward other answers to the Lord's question about his identity but Peter speaks for himself, prompted by God, and consequently he defines the fundamental belief of the Church.
This is quite different from the process that preceded the declarations of both the Immaculate Conception and even more so the Assumption, which where believed "always and everywhere" - even if in a slightly looser form than the formal 19th and 20th definitions. (The earlier disputes about the Immaculate Conception were more about "conception" and ensoulment than Mary's "Immaculateness", even when that was touched on it tended to dispute how it was achieved rather than the fact of it.)
Considering the degree of consultation with the bishops of the world that preceded both of these dogmatic definitions, even if no council was called, they seem really to be "concilliar" definitions, in the sense of being made not by the Roman Pontiff on his own but "with consent" of all the bishops of the world.
Some suggest that the Immaculate Conception and Assumption were used as ways of "demonstrating" Papal power or demonstrating Papal triumphalism, which may or may not have been their purpose but the effect was to introduce a degree of Concilliarism which reached its zenith in Vatican II.
Papal Infallibility seems more about defining disputed doctines which could divide the Church, than defining those on which everyone already agrees. The theological situation of the twentieth/ twenty-first century might suggest that Papal Infallibility is more concerned with holding Christians in unity, identifying where the Church is, over issues like sexuality - which includes women's ordination, which in the future will be the great divide amongst Christians.
For where Peter is, there is the Church and where is the Church there is Eternal Life.
Idle Speculations put up this video of the Declaration of the Assumption.