Ok so using your own logic, let's take both sides into consideration.

One side of the story says the summary was misleading. This side has nothing to lose or gain by lying. This is information from a letter written at the time the summary was released by the person in charge of the report the summary was about. 

The other side of the story says that the summary was not misleading. This side has nothing to lose by lying about this at this point since they have already made many public statements about the summary contradicting the information in the letter from mueller. This side has everything to gain by lying about this (if it stays in the darkness) because if the letter is true, it implicates Barr in possible criminal purjury charges (kinda a stretch) and possible ethics investigations. This sides information comes as a reaction from Barr's spokesperson today after contacted in response to the letter. 

Based on this information alone, an OBJECTIVE observer would conclude that while both sides are contradictory to atleast some degree, Mueller's letter would hold more weight. In this case, Mueller describes his concern with Barr's summary and his representation of it.