Cross Reference...I am having a little trouble fully understanding your OP. If what you mean is that the redemption of Christ is God "purifying a people for himself (tit 2:14)," than in a way I understand.
Redemption is that part of our salvation we had nothing to do with. It canceled out the penalty of Adam's transgression, [for which we are not guilty except by being a blood relation]. Redemption opens the way for us to have an 'advocate' with God, Christ Jesus, a blood relationship, something Adam never had.. We now are of a new creation if born again.
God "purifying a people" can only begin at one's new birth, not redemption, not at *salvation. At that point in time is he made a "child" of God and handed over to the teaching of the Holy Spirit for to become a son, one who has learned the ways of God, his Father per 1john2.. This is eternal LIFE as God planned it, set in Himself, for the new born of Him to enter into: "
And this is eternal life, that they [might] know you the only true God, and Jesus Christ whom you have sent." John 17:3 (ESV)
*salvation is one who is obviously legally saved but who has not yet entered into a new birth by Christ Jesus. He is given to
*"count the cost" before asking for the Holy Spirit.
*"For which of you, desiring to build a tower, does not first sit down and count the cost, whether he has enough to complete it?" Luke 14:28 (ESV) [This is the ultimate reason for salvation, i.e., that we become a son in whom the Father can be well pleased.]
Ask more if still not clear.