Benjamin accepts their declaration of a willingness to enter into a new covenant. He provides a little more definition of that covenant. They are no longer just the children of Israel, they are children of the Messiah. Their understanding of the future atoning mission has given them a larger view that the children of Israel, and, therefore, Benjamin, extends the covenant to cover this new understanding. They are spiritually born through their acceptance of the covenant, and the covenant requires understanding and acceptance of the future mission of the Messiah.
Just as seeing the word Christ in this verse tells us that it was translated in a Christian world, so may the presence of the word faith. That concept is elaborated in the New Testament, but translates the older understanding of allegiance, or perhaps faithfulness. Thus, they are changed through faith in that they have accepted, and declared allegiance to, a new covenant.
The two phrases in verse 8, “under this head,” and “no other name,” echo Benjamin in Mosiah 3:17, where he declared that there was “no other name given, nor any other way nor means whereby salvation can come.” Benjamin had earlier explained that there was only one other way, and now repeats that declaration as part of the acceptance of the covenant they have made.