This was His spiritual death. While being judged in our place, His humanity was separated from God the Father and God the Holy Spirit. It was His substitutionary spiritual death that was efficacious for our salvation.
We know that he was physically alive while being judged because He kept screaming, "My God (the Father), my God (the Holy Spirit), why hast thou forsaken me?" He was quoting Psalm 22:1 where the verb in the imperfect tense indicates that He shouted this over and over again. Christ was forsaken because " ...he(the Father)hath made him(Christ)to be sin for us, who knew no sin; that we might be made the righteousness of God in him"
When His spiritual death was complete, Jesus Christ shouted, "Tetelestai!" - the perfect tense meaning, "it is finished in the pasty with results that go on forever!"
Note that our Lord was still speaking after salvation was completed. Obviously He could not have spoken if He were physically dead! And certainly if He was still physically alive on the Cross after salvation was complete, His physical death could have nothing whatever to do with the payment for sin!
He died physically by His own volition- no one took His life! His work on the earth was finished, the Father's Plan called for Him to depart and He dismissed His own spirit.
source: The Blood of Christ by R.B.Thieme Pastor of Barachah church in Texas. Dallas Theological Seminary Summa Cum Laude. Nine years of Greek and five years of Hebrew. Now teaching isagogical,categorical and exegetical teaching of the Word of God.
Jesus is the perfect sacrifice prefigured in the Passover. There were four cups in the ritual. The third was the Todah (thanksgiving or Eucharista in Greek) offering of bread and wine. Jesus gave us His Body and His Blood as the passover lamb. In Jewish tradition, in the days of the Messiah, only the Todah offering would continue... and it does at every celebration of the Eucharist.
The fourth cup is the cup of consummation. Jesus, acting as priest and sacrifice on the cross completed the passover sacrifice with the drinking from the sponge. He then said, "It is finished." and gave up His Spirit in death.
John 19:30 Therefore when Jesus had received the sour wine, He said, " It is finished!" And He bowed His head and gave up His Spirit.
It is no more complicated than what is written in the text of Scripture. Your theologian went through some amazing theological gymnastics to avoid the 2,000 year old Catholic understanding... because obviously anything the Catholics believe must be wrong...
Maybe Jesus spoke those words (repeatedly? Where does the Bible say that?) because He had emptied Himself of His Divinity to experinence in His human nature the consequence of sin.