Save They Took Upon Them the Name of Christ

Read this verse carefully.  The clear implication is that the Nephites didn't take the name of Christ upon them at baptism, but that the taking of the name was a second step.  Moroni assures us that (in his day, presumably), the church only accepted new members who were serious; not only were they baptized, but they also took upon them the name of Christ (a covenant action that King Benjamin administered to his people at the temple).