The Old Testament prefigures the New. God in His infinite wisdom and mercy from all eternity already has a plan for mankind's redemption even before the Fall.

In 1 Corinthians 15:45, Paul compares Jesus to Adam: Thus it is written, "The first man Adam became a living being"; the last Adam became a life-giving spirit.” In Genesis 1, we learn that Adam was formed out of the dust of the earth by God. God breathed the breath of life into Adam. The perfect and sin-free Adam eventually fell in Genesis 3 by disobeying God, accepting the lies of the devil (about not dying and being like God), and eating the forbidden fruit from a tree, which allowed sin and damnation to enter the world. The perfect and sin-free Jesus, who said that he is the bridegroom of man in Matthew 9:15, also declared Himself to be The Bread of Life in John 6. He said that we must eat His flesh and drink His blood to have eternal life. This reference to eating, of course, is the antidote to overcome what Adam did. Jesus, who died on the tree of life known as the cross, commands us to eat the fruit from that tree (his flesh and blood), so as to give us eternal life and to overcome the two lies of the devil to Adam. The devil said that if you eat the forbidden fruit, “You shall not die.” (LIE!). By eating the fruit of the cross, the flesh and blood of Christ, “You shall live forever,” said Jesus (TRUTH!). The devil also told Adam that if he ate the forbidden fruit, he would be like God (LIE!). Jesus said that if we eat His flesh and drink his blood, He would abide in us, and we in Him (TRUTH!).

Similarly, Mary is called "the New Eve"by the early Church Fathers. Take, for example, Justin Martyr, who wrote within a couple of generations of the apostles. In his Dialogue with Trypho the Jew (ca. A.D. 150), Justin explains that Christ destroyed Satan’s work in the same way evil originally entered the world. Evil entered through Eve while she was still a virgin; so too salvation entered through Mary while she was still a virgin. Each woman willingly participated in the act they performed. Neither was an unconscious instrument. Eve listened to the serpent and conceived death. Mary listened to the angel Gabriel and conceived life. Justin sees this clearly in Luke 1:38 when Mary says, "Let it be to me according to your word." Thus, for Justin, Christ’s becoming a man involved his Mother’s willing cooperation in undoing the tangled web of sin that Eve introduced. Also St Irenaeus, the bishop of Lyons in the second century mentioned a similar thing. In Against Heresies, Irenaeus teaches that Christ embodied Adam and all his posterity in order to redeem mankind from sin. Basing his teaching on Paul’s inspired doctrine of Christ as the Last Adam (1 Cor. 15:45), Irenaeus viewed Jesus as reversing the effects of Adam’s sin by bringing the life and righteousness that Adam lost (Rom. 5:17, 18). Irenaeus saw the obvious implication. As Eve cooperated with Adam, the covenant head of humanity, so Mary cooperated with Jesus Christ, the covenant head of the new humanity. Thus Irenaeus says that Eve "by disobeying became the cause of death for herself and the whole human race, so also Mary . . . was obedient and became the cause of salvation for herself and the whole human race" (Against Heresies 3.22.4). Later he says of these two virgins, "Just as the human race was subject to death by a virgin, it was freed by a virgin, with the virginal disobedience balanced by virginal obedience" (ibid., 5.19.1).

You said "In Genesis 4:1, we see Eve had a child, and she thought the child was the Redeemer;
Eve knew that 'the woman' was meant to be her. She believed in what God said. So clearly, both Adam and Eve knew that God meant Eve by 'the woman'."

Probably. For Eve is not God, but if we see the harmony of Scripture in salvation history in both the Old and the New Testaments, very clearly the woman God refers to can only be Mary.