Samuel has identified the Messiah, now he identifies the role of the Messiah. This is the Atoning Messiah, not the Triumphant Messiah. The Nehorites might have continued in a belief in the Triumphant Messiah of the end times, as that role was certainly referenced by brass plate prophets. The conflict in Nephite society has always been this understanding of the Atoning Messiah, and the debates over the Messiah have revolved around his mortal mission, not his final mission at the end of time.
Samuel does not directly reference Nephi’s teachings to these same Nephites, but the intent of the two prophets is clearly parallel:
Helaman 8:15
15 And as many as should look upon that serpent should live, even so as many as should look upon the Son of God with faith, having a contrite spirit, might live, even unto that life which is eternal.
For Samuel, the phrase is “believe on his name… have a remission of [sins].” For Nephi, it is the Mosaic “look upon the Son of God…. [and] live.” In both cases, the prophet invokes the prophesied Messiah in his role as Redeemer. For both Samuel and Nephi, it is the redemption from sin and death that is the message. For both, the key is the belief in the Savior, either through the sacred acceptance of the name, or the symbolic “looking.”
Neither the name nor the serpent symbol is the effective elements. Both stand for our actions, not to dictate our actions. It is for this reason that both prophets also emphasize the necessity of our own repentance, explicitly in Samuel, and implied in Nephi’s “having a contrite spirit.”