Shalom. What Bible verses are used to substantiate this teaching, that when Adam sinned, his human spirit died? Aren't there several instances in the Bible where reference is made to the human spirit, as being alive and animate?
I do believe in the tri-partite nature of man: body, soul and spirit. However, the way I have been taught to believe is as follows.
As Peter mentions in his epistle that one day with the Lord is as a thousand years, and a thousand years as one day, so Adam died just short of a thousand years (930) and was buried, per the word of the Lord, who said: "For you are dust, and to dust you shall return." He died, therefore, in the same prophetic day in which he sinned, though not in the same 24-hour period, since by the mercy and forbearance of God his death was postponed by the blood of the creature that the Lord then skinned to clothe them - a picture of the atoning death of Christ.
Physical death, then, is the curse passed by sin to all mankind: we die and return to the dust. That is not to say that eternal death isn't a thing. For all those who die in their sin, outside the finished work of Christ, must still face eternal death = separation from God forever.
What then, occurs at the new birth, if not a resuscitated spirit? First, we receive the gift of the Holy Spirit as an "earnest" towards the day of redemption, who renews and revitalizes, yes, not just our human soul and spirit day by day, but ultimately our earthly body on the day of redemption. Anyway, that was my rant.
Soli Deo gloria.