If written out in full would this verse read "in the name of the Father, and in the name of the Son, and in the name of the Holy Spirit"? Or is there some other reason apart from trimming obviously implied words that the Father, the Son and the Holy Spirit [clearly 3 names] are said to have one name?
Wallace offers a very good explanation of the use of the term in the original language. It may help to understand exactly what is meant by the term
εἰς τὸ ὄνομα - into the name of.
I am not going to try to quote Wallace. I will just give the sense of his explanation. In the classical style of the first century language, the phrase "εἰς τὸ ὄνομα" - "into the name of" was often used as a legal term. This expression is found among ancient legal documents that recorded the transfer of property. If one purchased a section of land or a dwelling for example, a title transfer would be drawn up to show that this property was now εἰς τὸ ὄνομα - in the name of - the new owner. When Jesus commanded his disciples to preach the gospel to every creature and "baptize them into the name of the Father, Son, and the Holy Spirit," what he is commanding them to do is to baptize them into the possession of God. Christians are the objects of a property transfer - "out of the kingdom of darkness and into his marvelous light." Baptism then is a property transfer. This same language is used in Acts 2:38 when Peter commanded those present to be baptized ἐν τῷ ὀνόματι - into the name of - thus into the possession of Jesus Christ. This was for a two-fold purpose 1. For the removal of sin - Spiritual circumcision, Colossians 2:9-13. 2. To receive the gift of the Holy Spirit. He is the seal of the transfer of property, Ephesians 4:30, 1Corinthians 6:19-20. He is the seal of ownership to show that we have been bought with a price.
A Baptismal Formula
Since the phrase seems odd, some have questioned its authenticity as discussed here: Was the text of Matthew 28:19 changed?. This answer presents the evidence this text is original:
πορευθέντες οὖν μαθητεύσατε πάντα τὰ ἔθνη βαπτίζοντες αὐτοὺς εἰς τὸ ὄνομα τοῦ πατρὸς καὶ τοῦ υἱοῦ καὶ τοῦ ἁγίου πνεύματος [GNT]
Thus the OP's reading: "If written out in full would this verse read 'in the name of the Father, and in the name of the Son, and in the name of the Holy Spirit'" is incorrect. It should be:
Go therefore and make disciples of all nations, baptizing them in[a] the name of the Father and of the Son and of the Holy Spirit, [ESV]
[a] Matthew 28:19 Or into
The phrase may be confusing, because one expects either 3 names, or names plural, not name singular as in the actual text. But as the NET Bible notes, the intent is obvious (my emphasis added):
tc Although some scholars have denied that the trinitarian baptismal formula in the Great Commission was a part of the original text of Matthew, there is no ms support for their contention. F. C. Conybeare, “The Eusebian Form of the Text of Mt. 28:19,” ZNW 2 (1901): 275-88, based his view on a faulty reading of Eusebius’ quotations of this text. The shorter reading has also been accepted, on other grounds, by a few other scholars. For discussion (and refutation of the conjecture that removes this baptismal formula), see B. J. Hubbard, The Matthean Redaction of a Primitive Apostolic Commissioning (SBLDS 19), 163-64, 167-75; and Jane Schaberg, The Father, the Son, and the Holy Spirit (SBLDS 61), 27-29.
This is the Christian's trinitarian baptismal "formula," as Joseph Benson explains in his commentary:
Words which have been considered, in all ages of the Christian Church, as a most decisive proof of the doctrine of the Trinity; implying not only the proper personality and Deity of the Father, but also those of the Son and of the Holy Spirit. For it would be absurd to suppose that either a mere creature, or a mere quality, or mode of existence of the Deity, should be joined with the Father in the one name into which all Christians are baptized. “To be baptized into the name of any one implies a professed dependance on him, and devoted subjection to him; to be baptized, therefore, into the name of the Father, and of the Son, and of the Holy Ghost, implies a professed dependance on these three divine persons, jointly and equally, and a devoting of ourselves to them as worshippers and servants. This is proper and obvious, upon the supposition of the mysterious unity of three coequal persons in the unity of the Godhead; but not to be accounted for upon any other principles.” — Scott.
This idea of belonging to and becoming part of is also explained in Old Hermit's answer.
Also, it is incorrect to assume the terms τοῦ πατρὸς, τοῦ υἱοῦ, and τοῦ ἁγίου πνεύματος are themselves names. They are titles or positions. By way of comparison, consider "The God of Abraham, Isaac, and Jacob" which speaks of a singular God for three named individuals. This could also be stated as "The God of Sarah's husband, Rebekah's husband, and Leah's husband." Both phrases convey the same meaning.
Paul uses the baptismal formula when recalling the Exodus:
For I do not want you to be unaware, brothers, that our fathers were all under the cloud, and all passed through the sea, and all were baptized into Moses in the cloud and in the sea (1 Corinthians 10:1-2)
Matthew baptize into [the name] εἰς τὸ ὄνομα Corinthians baptized into Moses εἰς τὸν Μωϋσῆν
When the baptism is recorded, "Moses" is inserted, Thus in the instruction, "ὄνομα" functions like a "placeholder" to be filled by the specific name.
We can use a hypothetical example to illustrate how a singular name would be used for a three-fold entity. Suppose Jesus had said "Go therefore and make disciples of all Gentiles, baptizing them in the name of Abraham, Isaac, and Jacob." This instruction would be understood in terms of bringing Gentiles into Judaism by baptism (not circumcision) using the proscribed baptismal formula. That is, Jesus replaced a male only ritual with one which all people must undergo.
If we read that the Apostles went to the Gentiles, preached Jesus, and taught them to observe all that Jesus had commanded (Matthew 28:20), we would also expect to learn they baptized those who accepted. If we read, "So they were baptized into the name of Israel" we would know the Apostles understood the name of Abraham, Isaac, and Jacob is Israel. The baptism into a singular name brought them into the three-fold entity.
In reality, Jesus did not give an instruction to bring Gentiles into the Jewish faith, but to make children of God by believing in His Name (John 1:12):
Baptismal Formula Name Example: Baptize into the name of Abraham, Isaac, Jacob Israel Actual: Baptize into the name of the Father, the Son, and the Holy Spirit Jesus
1. NET Bible
The answer as to the one name of Matthew 28:19, I believe is seen best in the statement made in
1 John 5:7 For there are three that bear record in heaven, the Father, the Word, and the Holy Spirit: to and these three are one.
Yet if the originality of this text is in question as well, consider this most ancient context as conjuncted in John 1:1-14 as One name 3 functionaries, this can well be extracted from
Gen 1: 1-4 *1. In the beginning God... 2. And the Spirit of God moved... 3. And God said, Let there be...*
In the most ancient concept of Biblical writ we can see: God, His Spirit and His word which 'said...' , And in Genesis 2:4 the concept of YHWH / יהוה comes into being as derived from the concept of היה or הוה "was, and/or being" the name by which the Word speaks of him self as God and With God as moving by the spirit "... YHWH God made the earth and the heavens...".
This well represented in:
Psalm 33: 6 By the word of YHWH were the heavens made; and all the host of them by the spirit of his mouth.
The concept is literally seen when one speaks, you may see 'them' open their mouth but logic is being expressed by the breath or spirit (moving of void or air) which produces word.
I am not horribly fond of the term "Trinity", nevertheless this is an inescapable fact of scripture, that The Father by His spirit-breath speaks his Word to bring forth what has been preposed, and the Apostles and prophets bring it forth often in their expressions Mathew 28:19 is no exception, even if one would take the Majority Text approach in denial of it's originality to Mathew and only a Byzantium commentary, what damage does it do? None, if the sum of scripture proclaims the same message.
Therefore, if the book of Acts and other sittings say as in [Act 10:48 NASB] (48) And he ordered them to be baptized in the name of Jesus Christ... Is it not confirming that within that name of " Jesus Christ" is the name of the Father, Son And Spirit in the immersive work of Faith as commanded to be done in Baptism? Yes, there is no place in Scripture where the Son is mentioned without the Father's authority, and none where the Son speaks that is not by the Spirit.
I think is best said in 2Co 5:19 To wit, that God was in Christ, reconciling the world unto himself, not imputing their trespasses unto them; and hath committed unto us the word of reconciliation:
[2Co 5:18-19 ESV] (18) All this is from God, who through Christ reconciled us to himself and gave us the ministry of reconciliation; (19) that is, in Christ God was reconciling the world to himself, not counting their trespasses against them, and entrusting to us the message of reconciliation.
A sheriff could say "Open this door!" or "I'm arresting you!", and follow it with "in the name of the Law and of the King!".
He is not implying in any way that the Law and the King are the same thing, much less that they are a single person. He is simply stating the sources of his authority to force the door and make the arrest.
So when Jesus said "baptizing them in the name of the Father, and of the Son, and of the Holy Ghost", he was simply indicating by what authorities they would be performing the baptisms.