The “saints of the church” are the members of the church of God in the Nephite record — those who took upon themselves the name of Christ through baptism and were numbered among the people of God. Nephi saw in vision that the power of the Lamb descended upon the saints of the church of the Lamb, the covenant people scattered across the earth (1 Nephi 14:14).
After Alma baptized the believers at the waters of Mormon, King Mosiah granted him authority to establish churches throughout Zarahemla and to ordain priests and teachers. The believers assembled in seven churches, all counted as one church of God; those who desired to take upon them the name of Christ or of God joined, were called the people of God, and prospered in the land (Mosiah 25:18-24). During the Amlicite revolt, those who did not follow Amlici were called Nephites, or the people of God (Alma 2:11), and in battle the Nephites prayed to the Lord and were strengthened, so that the Lamanites and Amlicites fell before them (Alma 2:28).
The body of believers extended beyond one group or era. The Anti-Nephi-Lehies buried their weapons and covenanted never to shed blood again; when the Lamanites attacked, more than a thousand were slain without resistance, and a greater number than were killed joined the people of God (Alma 24:16-29).
Abinadi, expounding Isaiah, identified the seed of Christ as all who heard the holy prophets and looked to the Lord for redemption, naming them heirs of the kingdom of God (Mosiah 15:10-18). This new birth, the Lord told Enoch, comes by water, the Spirit, and cleansing through the blood of the Only Begotten (Moses 6:59). King Benjamin told those who entered the covenant that they were spiritually begotten as the children of Christ, his sons and daughters, and would be found at the right hand of God by the name they had taken upon them (Mosiah 5:7-11).