This story has been passed down throughout the generations, from mother to daughter. The OT as we know it, is the male version of things, written for other men. This is the female side of things, preserved through oral tradition, for other women. The version I heard from my own mother:
God created the species of adam (meaning "dirt"), both male and female. The female called herself Lilith, the male called himself Ish. Neither of them knew how to have a relationship, as they had no examples and such, and so they fought all the time.
Lilith got so mad at Ish one day after a fight over who would be 'on top' that she left him.
Ish was so upset over this, that he cried all the time and wasn't doing the job he was created for very well. God decided it was not good for Ish to be alone, so God decided to make another woman. Ish begged God to make the second woman out of one of his ribs, to make sure the woman would never leave his side. God told him that wouldn't work right, but removed one of Ish's ribs anyway, to show him... indeed Ish saw, and said that it was just a "bone of my bone, flesh of my flesh, it is a'isha, a piece of ish." God then returned the removed rib to dust... and proceeded to create the third woman.
This third Woman was created not from the dirt, nor from a part of Adam, but from divine inspiration and she would be ezer kenegdo (literally "equal to help from God") for Ish... and it was not Ish, but God who named her: Eve.
It was Eve who gave Ish the forbidden fruit... but it was not in disobedience of God, but rather, as God's instrument, as this was the only way that Ish could be given a conscience and begin practicing free will. Eve herself, already possessed this knowledge... she was afterall, an extension of God, and being so divinely inspirational, she didn't even have to lie to get Ish to take a bite of it.
After this incident, Ish and Eve were not kicked out of a garden, but rather, humankind was jarred out of ignorance which was bliss, east of eden (from the assyrian edeni, which means depression), when they were given control over their own lives, complete with responsibilities and consequences that come w/ it.
Since Lilith didn't have any forbidden fruit, she had no knowledge of the difference between good and evil, nor control over herself, running wild all over the earth, without conscience or shame... which is what humankind would be like today, if Ish had never taken the fruit from Eve.