This verse logically follows from the allegiances described in verse 12. Just as the sons of Amulon and the priests number themselves with the Nephites and accept dominion by the Nephite rulers, so did the Zarahemlaites. Mormon is reworking original texts dealing with Zarahemla’s political structures here, whether he is aware of the reason for including this passage or not. The significance of verses 12–13 is the subordination of potentially legitimate claims of rulership to the superior claim of the Nephite polity/lineage.
This is the context for “Nephite” here, rather than Jacob’s more general usage (Jacob 1:13), and it means the ruling lineage of Nephi. Both the children of Amulon and the Zarahemla Mulekites are declaring allegiance to the Nephite dynasty as represented by Mosiah1, Benjamin, and now Mosiah2. They are simultaneously giving up competing claims.