First, I’ve never formally been taught this eisegesis, so I cannot state with certainty that it as the doctrine of the Catholic Church. If there is error, it is mine. I am intentionally keeping this brief, which may add to error. I began this thought process after reading in the Catholic Doctrine of Atonement developed in the sixth Session of the Council of Trent, chapter ii. Which stated:
“Whence it came to pass, that the Heavenly Father, the Father of mercies and the God of all comfort (2 Corinthians 1, 3), when that blessed fullness of the time was come (Galatians 4:4) sent unto men Jesus Christ, His own Son who had been, both before the Law and during the time of the Law, to many of the holy fathers announced and promised, that He might both redeem the Jews, who were under the Law and that the Gentiles who followed not after justice might attain to justice and that all men might receive the adoption of sons. Him God had proposed as a propitiator, through faith in His blood (Romans 3:25), for our sins, and not for our sins only, but also for those of the whole world (I John ii, 2). “
This doctrine provided me with a context in which to read and understand many parts of Scripture, including some that I posted in this thread (Matthew 20:28, Mark 10:45, 1 Timothy 2:6). Throughout Pauline Scripture sin is represented as a state of bondage from which man is delivered by being redeemed, or bought with a price. "For you are bought with a great price" (1 Corinthians 6:20). " and “they sang a new song, saying, “Worthy are You to take the book and to break its seals; for You were (slain, and purchased for God with Your blood men from every tribe and tongue and people and nation.” (Revelation 5:9).
In the context of the Doctrine of Atonement, the Atonement is the deliverance from captivity by the payment of a ransom, a ransom that Jesus paid with His blood.
I think that ALL these terms, washing, redeeming and the rest are all images of the one thing. I think drawing “bright lines” between these aspects of the Atonement can tangle us up.
Satisfaction, expiatory sacrifice, conquering our ancient adversary, manifesting the truth of the Love of God — all these, in my alleged thought, are not different things but one Atonement understood from different approaches.
So here, blood pays the blood debt, there it covers our death-dealing sin with the blood which is life, over there it is a tide which sweeps death and Satan away, and then it is also the revelation of the depths of the Love of God in Jesus, a love so great that even after death blood and water are shed on us.
That's how I'd work it, anyway.