Well, the only mention of Khan's backstory in TWOK is when Chekov tells Terrell that Khan is a criminal and a product of late 20th century genetic engineering. Then Khan says Kirk stranded them on Ceti Alpha V.
As far as Harrison's motivations go, they may have been less recognizable than Khan's in TWOK (who was motivated only by pure, blinding revenge) because they were more subtle or nuanced (protect his commrades -- eventually free them, then go from there).
The scene with Spock Prime bothers me. I hope there's context to it. Why can't Spock find out about Khan from the ship's library computers, like in "Space Seed"? And from what I've heard (not having seen the movie yet), Spock Prime really doesn't say anything helpful.
Because Spock doesnīt need historical facts of who Khan was and what he did. He doesnīt trust Khan, who Kirk has teamed up with and space jumped into the USS Vengeance with. Spock thinks he needs a way to beat him in order to save the captain. Then, the logical thing is to call Spock Prime. He just canīt get the info he needs from the shipīs computer. Anyway, the whole conversation between them isnīt shown. So we have to imagine what Spock Prime tells Spock.
I was worried about this scene just being a fanwank, but it works well and feels natural in the movie. And, of course, it is great seeing Nimoy again