Okay, now this question about resurrections in the Bible has me wondering what the difference between resurrection and revival is. If resurrection "returns them to their proper embodied state, in which they will remain forever after", then how are the raising of Lazarus, the son of the Shunammite woman, the daughter of Jairus, and others, resurrections? Did these people become immortal (i.e. living forever, never dying) before Jesus did?
There are two types of Resurrections in the Bible.
1. Temporary Resurrection
Those who are brought back from the dead without a new body given to them. They will still die again when their time comes. This resurrection is mostly for the purpose of God's glory or to simply extend the life of the dead person on Earth as a result of prayer. This, even now is still happening in many parts of the world. Jesus resurrected Lazarus only to glorify God and himself through it.
2. Eternal Resurrection
Those who will be brought back from the dead for eternity. This will happen at "The Last Day". There are again two types based on the destiny of the resurrected person.
There are two cogent factors we must consider in answering your question;
The physical body cannot attain immortality;
After man had disobeyed God, he decreed that the human body would return to dust. Therefore the ultimate destiny of the physical body will be decay.
The Soul on the other hand is by nature immortal:
This living Soul is eternal (immortal) as is indicated by Jesus in:
Notice that Jesus said kill the body, but not kill the soul when talking about man, yet when talking about God he said destroy in Hell there is a very large difference there.
The original word used for kill in the Greek was:
While the word used for destroy was:
So in answer to your question what happened in these resurrections is that the immortal Soul, which came into being when life began, was returned from it's eternal habitation back into habitation in the human body where it originated. This is similar to God breathing the breath of life into Adam.
Once the Soul and subsequently the Spirit are created by God they will be eternal from then on which is the precept upon which Salvation is based in that you choose the eternal habitation for the Soul.
As far as revival goes it presumes that the Soul has never left the body to go to it's eternal habitation.
Resurrection means to be brought back to life after death.
Reincarnation means to be brought back to life in a new form after being dead. Those who have been reincarnated can die again and come back as some other living creature. Reincarnation is associated with some non-Christian religions that have this idea that people can be reincarnated as ants, worms or some other creature. This is why you'll see people who practice these religions not use fly spray or ant bait etc. This concept is not in the bible.
Lazarus was resurrected, he was brought back to life after death. But, Lazarus must have died again physically. Jesus died once, and was raised back to life and never died again. He concurred death.
As for being immortal, the worldly concept of immortality is to live forever in the body you have now. This is not what the bible teaches. The bible teaches that those who believe in Jesus and make him Lord in their life will live forever. Obviously not in their current bodies. Peter and all the apostles died physically, but their soul & spirit lives on. Lazarus would still have had to believe on Jesus to live forever despite being raised back to live in his restored body.
Physical death is when the spirit and the body are separated. When Stephen died, his spirit departed from his body and he was dead in a physical sense. But his spirit went on to be with God.
The other form of death is the spiritual one and this is what happened to Adam and Eve. Their spirits were separated from God.
True death is being separated from God eternally. The bible does not teach that death is ceasing to exist.