And those who did belong to the church were faithful; yea, all those who were true believers in Christ took upon them, gladly, the name of Christ, or Christians as they were called, because of their belief in Christ who should come.
As in the time of Captain Moroni (estimated to be about 70 B.C.), those who belong to the Mormon church today, who are true believers in Christ, gladly take upon themselves the name of Christ, or Christian! As stated in the prayer on the bread of the Sacrament of the Lord's Supper (the Eucharist or Holy Communion):
O God, the Eternal Father, we ask thee in the name of thy Son, Jesus Christ, to bless and sanctify this bread to the souls of all those who partake of it, that they may eat in remembrance of the body of thy Son, and witness unto thee, O God, the Eternal Father, that they are willing to take upon them the name of thy Son, and always remember him and keep his commandments which he has given them; that they may always have his Spirit to be with them. Amen. (D&C 20:77; emphasis mine.)