You are not correct. I just went and read the last part of Genesis Chapter 2 and Chapter 3 up to the verses in question.
The “woman” clearly refers to Eve.
I don’t know what you are pushing, but plain English is plain English.
You are putting meanings to the verses that are not there.
“You are not correct.”
Your opinion, but not that of the Universal Church, the Orthodox Church or even Martin Luther and many other proddies. You seem to have a very narrow, limited understanding of how the Old and New testament work together.