Both versions of the prophecy are right.

Rand, seeing the care shown and peace existing in Seanchan lands, will also remember that those who followed the Way of the Leaf served him in the Age of Legends, but now serve the Empress.

He will go to her, humbly, seeking a treaty. He will go alone.
He will have but one request-that he be allowed to give her the kiss of peace. In that moment, he will temporarily bind her to him as warder, so that she feels his pain, but also sees his sanity. He will swear to follow her if she swears to not leash any channeler allied to him. She will agree. He will then release the binding and swear to her.

Rand will then go to Egwene with his treaty, and be called insane. She will kick him out, furious. She will then strike at the Black Tower.

Rand will feel what is happening at the Black Tower, and arrive with Logain and forces allied to Fortuona to rescue Egwene and her Aes Sedai before Taim crushes them.

This sequence fulfills many prophecies and viewings, and fits with the increased focus on the Tuathan and Rands surrender of anger on Dragonmount.