What is the non-Catholic definition of papal supremacy? It is not in the language of the Catholic. We admit to papal primacy, but do not see it as supremacy. As far as I can tell, non-Catholics polemicists and apologists give “supremacy” the primary connotation of “dictatorship.” Let me propose my personal understanding of papal primacy, and I hope to get clarifications, additions, contradictions, discussion, etc. on the issue. Understand that this is my personal understanding.
Papal primacy means that the Pope is First Among Equals. He is equal to all other bishops as a bishop; all bishops are equal to him as bishop. He is equal to all other Patriarchs as a Patriarch; all other Patriarchs are equal to him as Patriarch. In these categories, he has primacy of honor only, and his prerogatives are the same as any other bishop or Patriarch in their particular jurisdictions. These common prerogatives are obtained de jure divino and are not from the consent of any other bishop or Patriarch with respect to other bishops or Patriarchs. However, Patriarchs do have a disciplinary prerogative over those bishops under them in order to ensure the unity of faith.
The Pope has another prerogative that no other bishop has. This prerogative is exercised in extenuating circumstances for the good of the Church. And this prerogative, like all other prerogatives of bishops, but in this instance particular to the First Among Equals, is obtained de jure divino, not from the consent of any other bishop or Patriarch. The nature of this prerogative is to ensure the unity of the faith universally, where the Patriarch does so in his jurisdiction, a bishop in his diocesan jurisdiction, and a priest in his parish jurisdiction. In cases of doctrine and morals, for the unity of the faith, the Pope can juridically impose a sentence, and his sentence will be infallible, and this not by consent or approval of the rest of the Church, but by the special provenance of God for the Petrine office. The sentence will be theologically infallible without the consent of the Church, but for the sake of the unity of the Church, the sentence will need the agreement of all the other bishops. No doubt, the Pope exercises this prerogative only as circumstances permit or dictate, and not ordinarily (that is, it is to be exercised extraordinarily). This prerogative is considered “ordinary” not because it can be exercised whenever he chooses (since it should be exercised extraordinarily), but simply because it is directly from God and not from the consent of the Church. The common prerogatives of all other bishops are regarded “ordinary” for the same reason (as well as the fact that they can so choose to exercise it whenever they wish, a factor missing from the definition of the “ordinary and universal papal jurisdiction”). In all circumstances, the Pope is bounded in his decisions by Sacred Tradition.