It would go a LONG way to clearing up mistaken ideas about Papal Infallibility, if Catholics (or any Christian) would understand that it is the CHURCH which is Infallible. (Church Infallibility.) This was recognized from Apostolic Times because Jesus is our Head, our King, and our High Priest. Jesus has sent the Holy Spirit to be with His Church and guide it. This is what accounts for Church Infallibility. It was always and everywhere held by Christians because the Church is a Divine Institution.
Theologically speaking HOW infallibility is exercised in and by the Church, was articulated over time. Because infallibility refers to teaching, it is attached in a particular way to the Magisterium. The Magisterium, in a Council, when in communion with the Pope is certainly another way. Vatican I, after long speculation and generally held belief included the Pope as having an individual exercise of Church Infallibility. After Vatican II, even the LAITY (or more correctly “the Faithful”) were included in this exercise of Church Infallibility, when in union with the Magisterium and the Pope.