Clearly punching Dan in the nose constituted a battery. However, Pete can claim that he was acting in self-defense when he struck Dan. The right to self-defense arises when a person reasonably believes that he is about to be battered by another. When Dan grabbed Pete s arm, Pete knew he was being battered. Pete also knew that he had stolen nothing.
Reasonable mistake is an excuse in self-defense.
Pete vs. Dan battery. Dan can claim merchant s privilege, possibly, for the battery. When their reason for this belief comes form a third party, the source has to be one which a reasonable person would rely on given the circumstances. However, not all jurisdictions allow the use of force with the merchants privilege. If Dan is in a jurisdiction that does allow force, he was probably reasonable in its use by grabbing Pete s arm, and would not be liable for battery.