The decisive legislation on this matter was promulgated at the Second Ecumenical Council (A.D. 381) in its 7th Canon: “Those heretics who come over to Orthodoxy and to the society of those who are saved we receive according to the prescribed rite and custom: we receive Arians, Macedonians, Novatianists who call themselves ‘pure and better,’ Quatrodecimans, otherwise known as Tetradites, as well as Appolinarians on condition that they offer libelli (i.e., recantations in writing) and anathematize every heresy that does not hold the same beliefs as the holy, catholic and apostolic Church of God, and then they should be marked with the seal, that is, anointed with chrism on the forehead, eyes, nostrils, mouth and ears. And as they are marked with the seal, we say, ‘seal of the gift of the Holy Spirit.’ As for Eunomians, however, who are baptized with a single immersion, Montanists, who are called Phrygians, and the Sabellians, who teach that Father and Son are the same person, and who commit other abominable things, and [those belonging to] any other heresies - for there are many of them here, especially among the people coming from the country of the Galatians, - all of them that want to adhere to Orthodoxy we are willing to accept as Greeks *. Accordingly, on the first day we make them Christians; on the second day, catechumens; then, on the third day, we exorcise them with the act of blowing thrice into their face and into their ears; and thus we do catechize them, and we make them tarry a while in the church and listen the Scriptures; and then we baptize them.”
In this way the Holy Church made the rules: by what order to receive those who come into Orthodoxy from heresy. Those who have a correct baptism are received without re-baptism. Those who do not have baptism in the name of the Holy Trinity - are received by way of Baptism. It must be noted that the Arians and Macedonians held to a wrong teaching about the Persons of the Holy Trinity, but the actual faith in the Holy Trinity, in the Father, Son, and Holy Spirit, was there, and this was sufficient, in the opinion of the holy Church for recognizing the validity (sufficiency) of their baptism.
- The same goes for SDA today.
In addition, the OP questions:
- LDS are baptized with a Trinitarian formula, but lack an actual faith in the Holy Trinity. Rather naming three Gods, in which they believe, with familiar Christian names. But holding to a faith that these three are among many gods, including a goddess mother, and one day themselves as among this group of gods.
-Christian Scientists don’t have a baptismal rite. They view baptism as a form of daily purification.*
And I suppose the same distinction applies to Oneness Pentecostals as opposed to Pentecostals who are Trinitarian? Thanks for the information.