Huh? I don't follow your logic on any of that. And no, I don't have it backwards. No one comes to Jesus unless the Father draw him. God sends his Spirit to convict you of your fallen nature, invite you in, etc. Then, if you receive it, you repent of what you are and get cleansed in baptism, then Holy Spirit becomes a part of your very soul.
Add: I suddenly perceive that perhaps you are trying to take the metaphor too far. I don't know, but I do know what I'm talking about whether I know how it does or doesn't fit with the metaphor to the extent you are looking for.
Or you're not taking the simile far enough, and this is a problem pandemic proportions within the Church. Most people tend to gloss over similes, such as spiritual death, sheep, new birth, etc. , and in so doing destroy the analogies to the point where there is barely any resemblance between the two kinds of objects (physical and spiritual).
For example, what you describe in your first paragraph, earlier you characterized all those actions (drawing, convicting, etc.) as "external" activities of the Holy Spirit; whereas I see those kinds of things as being
internal signs of life within a soul that is bringing the soul to the point of it's "new birth". In other words, I see the "new birth" as a process of Regeneration, just like physical birth is a process of pregnancy that began at conception. This spiritual birth, therefore, takes when a person bears witness to the world that s/he is a believer which is when that person becomes conscious and aware of God's work within and acts accordingly. Likewise, the world knows that a miracle of physical birth takes when the child is born -- the newborn being the testimony of this miracle, etc.
And most importantly, my understanding of Regeneration and the New Birth harmonizes very nicely with the terms of the unconditional New Covenant in which God promises to act unilaterally upon his chosen, covenant people. For example, no one comes physically into this world by an act of his own will -- but strictly by God's eternal decree. The same goes for the elect's spiritual birth.