But what is the source of this definition? They don't give a biblical reference, they just expect you to accept it. The bible doesn't distinguish between Adan's "physical death" and his "spiritual death". Paul doesn't either, at least not specifically. Neither does anyone say that death is a separation. It might be hinted at, but it's not clearly defined that way in the Bible.
It is found in Genesis 3 when Adam disobeyed God. The robe of white was removed if you keep the same term from Revelation 6:9-11
9 When the Lamb broke the fifth seal, I saw underneath the altar the souls of those who had been put to death for proclaiming the Word of God, that is, for bearing witness.
10 They cried out in a loud voice, “Sovereign Ruler,
HaKadosh, the True One, how long will it be before you judge the people living on earth and avenge our blood?”
11 Each of them was given a white robe; and they were told to wait a little longer, until the full number of their fellow-servants should be reached, of their brothers who would be killed, just as they had been.
When this robe was removed Adam and Eve were ashamed, naked, and hid from God. From Adam to the 5th seal, very soon in the future, no human could put the robe of white on. That is our spirit. There is no verse about a spirit that dies. That is why the spirit has been with God. If there are verses about a spirit, you have to use context. The spirit was the whole image of God. Genesis 5:3
3 And Adam lived an hundred and thirty years, and begat a son in his own likeness, and after his image; and called his name Seth:
We see that the first son Adam had after loosing God's image, was in Adam's own image. Spiritual death is not only separation from God, it is separation of this robe of white, that is the image of God.
On the mount of transfiguration do we see the affects, Luke 9:28-31
28 And it came to pass about an eight days after these sayings, he took Peter and John and James, and went up into a mountain to pray.
29 And as he prayed, the fashion of his countenance was altered, and his raiment was white and glistering.
30 And, behold, there talked with him two men, which were Moses and Elias:
31 Who appeared in glory, and spake of his decease which he should accomplish at Jerusalem.
The spirit is with God, because that is our glorified image from God. It is spirit, because God sends His own Holy Spirit as a replacement. The controversy of the start of a human seems not settled. Psalms 139:13-16
13 For you fashioned my inmost being,
you knit me together in my mother’s womb.
14 I thank you because I am awesomely made, wonderfully; your works are wonders — I know this very well.
15 My bones were not hidden from you
when I was being made in secret,
intricately woven in the depths of the earth.
16 Your eyes could see me as an embryo,
but in your book all my days were already written; my days had been shaped
before any of them existed.
Even the psalmist understood that as an embryo God was in control. God sealed us with His Holy Spirit when we ourselves were born as sinners. God already knew our whole life. Our spirit was witness in heaven as well to all that we are. God could use any means to directly communicate with us. I think it is through our spirit. Romans 8:15-23
15 For ye have not received the spirit of bondage again to fear; but ye have received the Spirit of adoption, whereby we cry, Abba, Father.
16 The Spirit itself beareth witness with our spirit, that we are the children of God:
17 And if children, then heirs; heirs of God, and joint-heirs with Christ; if so be that we suffer with him, that we may be also glorified together.
18 For I reckon that the sufferings of this present time are not worthy to be compared with the glory which shall be revealed in us.
19 For the earnest expectation of the creature waiteth for the manifestation of the sons of God.
20 For the creature was made subject to vanity, not willingly, but by reason of him who hath subjected the same in hope,
21 Because the creature itself also shall be delivered from the bondage of corruption into the glorious liberty of the children of God.
22 For we know that the whole creation groaneth and travaileth in pain together until now.
23 And not only they, but ourselves also, which have the firstfruits of the Spirit, even we ourselves groan within ourselves, waiting for the adoption, to wit, the redemption of our body.
We see that the two spirits are in communication. Verse 19-21 says we long to be sons of God in our full form. Not only is the body redeemed, but the soul itself will be joined by our glorified spirit.
Now we know this fleshly body cannot enter heaven or Paradise. It will return to dust. But Paul claims our soul will also be changed when the robe of white is put on. We shall all be changed. The dead first in the 5th seal. Then when the Lamb appears in the 6th seal, those alive in Christ will receive our robe of white (glorified bodies).
The seals opened the Lamb's book of life. The book provided by the Atonement on the Cross, that was written before the foundation of the world. As stated, the OT saints could not enter Paradise until the cross. They were in Abraham's bosom. But at the Atonement, they were set free from the sting of death. They now make up those souls under the altar.