What does "in the name of" mean in Baptism rituals?

"Matthew 28:18 is an authority saying which appeals to the solemn onoma (“name”) of Christ (“in my name”)…This authority saying is probably a further expansion of Luke 10:19 (= Psa. 91:13)…Compare Matt. 4:6 and parallel…R. Bultmann [supposed that] the missionary charge did not contain a triadic onoma but a Christological “into my name.”

The “name” stands for the person himself, recalls the appeal, proclamation, instruction, confession of someone…It seems to me necessary to retain this “baptismal statement” whatever our reservations. The short form found in Eusebius’ ante-Nicene writings can hardly be regarded as original…There is no avoiding seeing this “name” that is transmitted in triadic form as a normal confessional formula at the time of Matthew and the Didache…In Matthew “teaching” gains its own emphatic significance. The words which Jesus transmitted to the disciples are now his real legacy, and it becomes the task of the disciples to preserve and hand on the tradition in the form of the “commands”…Matthew thus underlines the special significance of the sayings tradition and reminds us of the related material of the Johannine farewell discourses which can likewise set the words of Jesus under the rubric “commands” (John 13:14; 14:15, 21; 15:10, 20)…This transmission is wholly unecstatic. It does not make a gift of the spirit central, though the triadic “name” includes a liturgical echo of the gift of the spirit.

[n. 19: It is an important fact that Matthew 28:19 and Didache 7:1, 3 place the triadic formula under a single “name” (“into the name…”). Triadic formulae are popular in Hellenistic antiquity, but require a thorough theological interpretation. This threefold “name” in Matthew clearly aims to summarize the peculiarity of early Christian proclamation as against other baptismal formulae and cannot be taken in the sense of later Trinitarian doctrine. Cf. E. Norden, Agnostos theos (1913), 348ff.]"

Otto Michel, The Interpretation of Matthew, “The Conclusion of Matthew’s Gospel,” pp. 35-41.

“Baptism is not a distinctive or uniquely Christian idea [other] examples include the Hindu rituals in the Ganges River, the purification ritual in the Babylonian cult of Enki, and the Egyptian practices of purifying newborn children and the symbolic revivification rites performed on the dead. Baptizo and related terms were used to define ritual practices in early Cretan, Thracian religions & Eleusinian mystery religions and in several Gnostic groups and cults. 

[In Judaism] the ritual of washing was similar to baptism in its purifying implications (Mk 7.4; Heb 9.20)…it is natural to seek a prototype for John’s baptism within Judaism of the first century. But determining the relationship between Jewish practices And understandings of baptism or lustration and those of John [the Baptist] or the early church is fraught with difficulties…Jewish proselyte baptism served to cleans the convert from moral and cultic impurity. John’s baptism for the remission of sins reflects a similar concept. 

There is, however, no clear evidence prior to A.D. 70 that proselytes underwent baptism as a requirement of conversion. This has been argued forcefully, in spite of the continued citation of [extra-biblical texts]…There is no mention of proselyte baptism in the OT, Philo or Josephus…It is therefore doubtful that proselyte baptism existed, at least as a clearly analogous rite, in John’s time. 

It is especially significant that John’s baptism was received by Jews rather than Gentiles. John demanded moral repentance and cleansing for the Pharisees. Jewish proselyte baptism, of course, served as an initiation rite for Gentiles, but Jews, since they were already the people of God, did not need the rite. If John’s baptism was developed from Jewish proselyte practices, he transformed it significantly.”
J.D.G. Dunn, Baptism in the Holy Spirit.

“It is still a matter of dispute whether [Jewish proselyte baptism] was in existence at the time of John the Baptist or not…It is somewhat singular that no baptism of proselytes is forthcoming until about the beginning of the 3rd century; and yet no competent scholar doubts its existence. Schurer is full of contempt for those who insist on the argument from silence.” 

International Standard Bible Encyclopedia, vol. 1, ed. Barr.