The way I see it, Clive doesn't see himself as much more than a sacrificial pawn to achieve a greater goal and he uses the names of those who he believes are the true inspirations behind his goal. He is completing Cid's goal, so he takes Cid's name. Joshua punches him in the face later in the game because he realizes what Clive is doing and makes him start thinking more introspectively. This ties very well into the book, because it would mean that Clive believes that this was truly Joshua's story, and as his Shield, he was the executor of his ultimate goal.
Joshua's goal, meanwhile, was to be Clive's Shield. A goal he fulfills in the end, by giving him Phoenix to defeat Ultima. Which in itself is directly a reference to the book. Clive became the Phoenix in the end, something Joshua always wanted. It would only be fitting and a fair honor to his brother to pen the book in his name, given he is now, ultimately, what his brother always represented. Even if he no longer has that power.