It sure seemed like he needed to prove himself, but then on top of that, you are right, MrBob, he should not have argued with Caspian in front of the rest of the soldiers... If he had to argue with Caspian, do it privately. But they did not have to argue at all, they should have shown move respect for each other.
I think that Peter should not have gotten into any fights and the fact that he did just shows how bad his pride was. His pride should not have been in that much control over him. He should have known that no one would respect him in England.. No one would know who he was, and if he told them, they would laugh at him and call him crazy.. When he got to Narnia, not only did no one believe who he was, but to top that, not many people respected him as they should have.. Yes, his attitude was not the best, and I do not like that Disney had to make him act that way, but I do realize why he did what he did, and that a lot of it was, yes, pride. But a lot of it was that he was trying to help Narnia in every way the boy he had become could help Narnia...
Caspian should not have been so selfish. When he learned about Miraz murdering his father, he wanted Miraz dead and he wanted it done the way he wanted it done, basically, he wanted to do it himself.. He was selfish in that respect. Then, Nikabrick was able to appeal to that selfishness by telling Caspian that the werewolf and hag would somehow destroy Miraz. It was Caspians fault that the witch almost came back, because he was the one that allowed her to be brought back in the way she was in the first place. Peter was trying to stop what was happening with the White Witch, but stepped into a magical circle, which he probably did not know was magic. Caspian was the one who allowed the circle to even be made and the witch to even have come back like that, none of that was Peter's fault... They both change by the end, after the witch almost coming back, actually... that was good, that was what I enjoyed... Well, part of what I liked anyways...