The Catholic Church of Christ believes the word “Baptism” (capitalized as a sacrament) represents a specific sacrament, and the Protestant Church of Christ believes the word—capitalized or not—represents a specific sacrament (so to speak) as well as a specific mode: immersion, and only immersion. The Catholic Church and the Protestant group agree that Baptism by immersion is valid, but the Catholic Church has always understood that Baptism is not relegated to its alleged pheno-linguistic parameters, but as a sacrament, includes pouring as a proper mode (often referred to as “sprinkling” by non-Catholics, but is in fact a pouring action when performed most properly).1
It is with great pleasure that the Catholic Church of Christ recognizes the validity of your group’s “first” Baptisms (there are no “re-Baptisms”; cf. Ephesians 4:5), and they are valid because they are still within the scope of the Catholic Church’s teachings (the Church is the foundation and conduit of truth: cf. John 14:16-18,26; 16:13; Ephesians 3:10; 1 Timothy 3:15); with water, in the name of the Father, the Son, and the Holy Spirit, and for the forgiveness of sins (regeneration). The difference, however, is that your group has deliberately narrowed its understanding of acceptable modes—an evolution that is tied to a new (= Protestant) belief that infants should not be baptized, which I will address in the next chapter.
In addition to your erroneous assumption that the Bible is intended to provide a thorough exposition of a proper mode (the Scriptures do not, and cannot, claim to be a “sole authority” or thorough expounder of the subject), your group’s confusion seems to largely stem from four false premises. It goes without mentioning, though I must, that not all members of your group accept every false premise as true, but all members do accept some of them in varying degrees. I will show how your group’s premises are false and simultaneously show you how the Catholic Church of Christ’s position is more reasonable.
1 “Sprinkling”, as a mode, does not affect validity, only licitness (strict adherence to law). Catechism of the Catholic Church #1239: “Baptism is performed in the most expressive way by triple immersion in the baptismal water. However, from ancient times it has also been able to be conferred by pouring the water three times over the candidate’s head.” Code of Canon Law, 854: “Baptism is to be conferred either by immersion or by pouring.”