I suggest that you read Pope Benedict’s “Spirit of the Liturgy”
First of all, the sacrifices being offered by the Minsterial Priest and the Common Priest are different.
The English translation being used now does not reflect that distinction, which is present in Latin, and as mentioned above, in the new translation.
The Ministerial Priest, and he alone, offers the Eucharist. We, the Common Priests, offer ourselves, our obedience and our good works along with, but distinct from, the Sacrifice of the Altar.
A Priest may only sacrifice what he is in possession of. The Minsiterial priest, as persona Christi, IS Christ and may therefore offer the person of Christ. We are not Christ, so we may not offer Him.
But we are creations of God and inspired by Him to do His Will on Earth, so we are to offer that ourselves, our bodies and souls and all the faculties thereof; the simple offering we can.
So, in reference to the original question, the possessive nouns differ because the translation stinks and it’s in the process of being corrected.