The Consilium was clear about the need to educate the laity on the “modes” by which we encounter the Real Presence of Christ. Clearly, the thinking in the sixties was that an expanded view of this would assist the laity in a deeper understanding and appreciation of God’s activity and presence in the Church and the world.
9. In order to achieve a deeper understanding of the eucharistic mystery, the faithful should be instructed in the principal modes by which the Lord is present to his Church in liturgical celebrations. [See Sacrosanctum Concilium art. 7.]
He is always present in an assembly of the faithful gathered in his name (see Mt 18:20). He is also present in his word, for it is he who is speaking as the sacred Scriptures are read in the Church.
In the eucharistic sacrifice he is present both in the person of the minister, “the same now offering through the ministry of the priest who formerly offered himself on the cross,” [Council of Trent, Decr. de Missa cap. 2: Denz-Schon 1743.] and above all under the eucharistic elements. [See Sacrosanctum Concilium art. 7.] For in that sacrament, in a unique way, Christ is present, whole and entire, God and man, substantially and continuously. This presence of Christ under the elements “is called the real presence not to exclude the other kinds, as though they were not real, but because it is real par excellence.” [Paul VI, Encycl. Mysterium fidei.]
It’s a “both/and” thing. “Both/and” has been the only way I’ve ever heard it presented.