I have a problem with every single one of your points.
1) I know what the Human-Forerunner war is. Guilty Spark says to Chief, "You are the child of my makers. Inheritor of all they left behind. You are Forerunner."
2) There is a BIG difference between a conscience and a consciousness. I think you confused them. Consciousness is more like self-awareness.
3) If it had been talking about absorbing him into a Gravemind, he would be destroying "breath, shape, and form" He claims to be a Precrursor, those who made the Forerunners.
4) I believe that the Precursor threw away this belief after the evil rebellion of the Forerunners. "Our answer is at hand." "Child of my enemy... I offer no forgiveness." The enemy is Forerunner.
5) He refers to the enemy as Forerunners, oh I don't know, maybe because they killed off the Precursors and also created the Halo Rings and blew most of the galaxy to Hell to kill the Flood.