I think here are a few mistakes, but you are certainly partly right. There is a wealth of evidence that points to the reality of the symbolic story in Genesis.

Firstly, it's necessary to realize that the fruit is symbolic. The idea that God would put a literal poisonous fruit in the garden in order to test his child, and that if they failed the test the result would be death, is frankly ludicrous. if any parent here today did this, we would put in prison as a deranged murderer. That's not love. Also, would a loving parent carelessly place a most tempting fruit in the middle of the garden that HE created? No. Also, Jesus himself said "Not what goes into the mouth defiles a man, but what comes out of the mouth, this defiles a man." Matt 15:11. So how could a literal fruit cause human beings to become corrupted?

Moreover, the sin of Adam and Eve is passed down, generation to generation. A food or fruit may hurt the person it eats, but that toxicity would not be passed down through each generation. No, what is inherited down through generations is passed down through lineage.

As you say, Adam and Eve were warned they would die when they ate the fruit. there is NO fruit in existance that any of us would eat if we knew for a certainty that we would die. Adam and Eve were not starving. So, it's only logical and sensible that the fruit represents something that was so tempting, so desirous, that Adam and Eve feared even death.

So what is the "fruit" a symbol of? First, consider the two "trees" in the garden. One, the Tree of Life, and the other the Tree of the Knowledge of Good and Evil. What does the Tree of Life symbolize? In the old testament: Hope deferred makes the heart sick, but a desire fulfilled is a tree of life. Prov. 13:12. New Testament: Blessed are those who wash their robes, that they may have the right to the tree of life and that they may enter the city by the gates. Rev. 22:14

What did the Israelites hope for? What about Christians since the time of Jesus? if the hope of of humanity is the Tree of Life, then what was the hope of Adam? When he fell, God 'blocked the way to the tree of life'. What did Adam hope for? Adam hoped to reach perfection, oneness with God, his creator. But he fell, and this became impossible. The Israelite hoped to welcome and unite with the messiah, Jesus. And the greatest hope of all Christians is to welcome and unite with Christ at the second coming. Christ, the perfection of man, the True Adam.

If the Tree of Life symbolizes perfected Adam in the garden, what about the tree of K of G&E? What is fruit? It contains the seed for the next generation. In many, many cases, the bible uses the expression "to know" to mean 'unite with sexually'. ("Adam knew his wife...." Gen 4:1)

What lies at the very center of God's vision and ideal for his creation? It is his children, humanity. Adam and Eve. So it makes sense that the two trees in the center of the garden actually symbolize Adam (or Adam's potential to become perfect) and Eve. The 'fruit' of the T of KGE is the symbol for "Eve's love". It was Eve's sexual love that held the seed for the next generation.

So what was the commandment? The Command eas necessary because in an immature state, it might be possible for the power of love to overwhelm the power of truth. Once Adam and eve's love was matured into unity with God, and unity with the truth (to become the Word) there would be no possibility they could fall. But in an immature state, of the power of love came to them from a non-truth direction, it could derail them. This is why God gave the command to Adam (and him to Eve). To prevent them from falling to the power of love.

Lucifer knew this. And, he also knew that if Adam reached perfection - became perfected Adam - then Adam would be king of the universe. But up until Adam and Eve were created, Lucifer was extremely powerful, and had a monopoly on God's love. But when Lucifer thought about Adam becoming MORE beloved of God than himself, he felt jealousy. That was a natural outcome of his nature to desire love. But because God knew this, he gave the command to Adam and Eve to prevent them from falling to Lucifer's approaches. Then, if they resisted properly, through faith in God's command, they would mature, fulfill the vision of God's creation, and Lucifer's wayward leanings would no longer be a problem, because the power of love perfected in a human being is as great as God's own love itself.

(Adam failed in this, and that is why Jesus had to come, to restore Adam's mistake, and become, 'the second adam' and 'the true adam'..)

Lucifer understood that if he could gain control over Eve though an unrightous love, a love without God at the center, then he could also gain control over Adam, and he thought in this way, he could forever keep his position at the top of the creation food chain. Lucifer was motivated by jealousy of Adam, and as he got closer and closer to Eve, and spent more time with her, eventually lied to Eve and tempted her to unite with him in love.

But its critical to recognize that Lucifer was an angel, a spiritual being. So Lucifer's uniting with Eve in fallen love was a spiritual union, not a physical one. However, once Eve fell with Lucifer spiritually, she then went and tempted Adam and they fell through their physical union. So that fall of man happened in two stages: the spiritual fall and the physical fall.

Because the Angel does not have a physical body, there could be no offspring from the spiritual union with Eve.

but later, Adam and Eve began to live together as man and wife - under Lucifer - Satan's dominion, not God's - and they produced children. Cain REPRESENTED the first fall, the fall in the spirit, which is further from God's truth and that second fall, the fall in the flesh. Luficer was NEVER meant to be Eve's partner in sexual union. Totally, 100 wrong. But Adam WAS original meant to be Eve's husband, so their union, although fallen, is LESS wrong that the union between Lucifer and Eve.

This is WHY the firstborn in scripture symbolizes evil and the second born symbolizes good. Both are relative, but until the arrival of Jesus, ALL humanity, both abel and cain, etc, are all in Satan's lineage.

This is ALSO why we must be reborn through Christ. We must be reborn because we are first born of the fallen lineage. Christ is thus the true Adam, and the true ancestor. When we unite with Jesus and receive the Holy Spirit, we are reborn into his lineage spiritually. However, because our flesh still belongs to Satan's lineage, Christ must come again to give rebirth to humankind physically, in the flesh.

The key to understand here is that the so-called 'lineage of Satan' that you describe is all who have not been yet reborn through Christ.

The conflict in History is indeed conflict between the lineages that symbolize relative evil and those that symbolize relative good. How is the conflict resolved? When the representatives of evil, symbolizing Lucifer (or the first born), can submit and humble themselves to the representatives of good, symbolizing Adam (aka the second born) then God claims that re-union and it reverses the original fall in the garden.

This is exactly why Jacob became "Israel" and the founder, with Abraham and Isaac, of the chosen people. Jacob, through many trials and tribulations, was the first person in biblical history to gain victory over "the first born" - his brother Esau. When Esau humbled himself to Jacob, Jacob could stand separated from Lucifer's accusation and claim, and this became the foundation for the lineage that would eventually give birth to Jesus, the True Adam.

In the chart, what you refer to as Yahweh's children and Satan's children are relatives positions. And, this has to do not so much with DNA, but with who has claim over that lineage or not.

For time, this is a more accurate understanding of the essential core.