In the first, Peter's authority is established in his testimony, which came via a revelation of Truth from our Heavenly Father. In the second, Peter's authority was absent, because he was thinking "not as God does, but as human beings do." Thus he is described contrastingly, as a "rock" (as a building-block for erecting the Church) and as a "stumbling stone" --an obstacle to the Lord.
Thus the authority, and the contrasting limit, are there from the Lord's own words in the Gospel.
Over the years, people's estimation of this authority varied almost, you could say, "from Zero to Hero". There needed, eventually , to be a clarifying definition.
All who are familiar with Church history know that even the most ancient, Biblically-consistent beliefs are not generally formally defined until some crisis forces the issue. Unless there are differences of opinion or interpretation which put the unity and peace of the Church in peril, the Pope doesn't go around issuing definitions proactively.
This has been true even of sound Apostolic Christology, which was believed from the days of the Gospels, but not defined until the Arian heresy arose in the 4th century, which erroneously asserted that Our Lord the Second Person of the Blessed Trinity was not "always" God, that is, not "co-eternal" with the Father and the Holy Spirit. That had to be solidly, precisely and definitively settled, and hence the Council of Nicaea and subsequent councils.
So Christ being "True God and True Man," One Divine Person consubstantial ("one in being") with the Father, having two natures, human and divine --- that was known from the days of Peter, James and John. It as always a dogma of Christianitym 1st century. But it was not formally "defined" until the 4th century, and then under the impetus of the spreading Arian heresy. The Nicene definition was settled by the Bishops and received/ratified by the Pope.
As to the pope's Petrine authority -- his infallibility --- it was defined in 1870 about as minimally as it is logically possible to do so.
It has no "positive" content. It does not say that the Pope is inspired as were the human authors of the Bible or the writers of the Gospels or Epistles. It does not say he will teach in the most effective way, or the most timely way, or the most persuasive way. It only states what he cannot issue error concerning the Deposit of Faith (Faith or Morals) in such a way as to be binding, "de fide," on the whole Church, for everyone, for all time.
Please provide your source for this. You don’t write this way.