Father, Son, and Holy Ghost are titles that speak of one Person, Jesus Christ; and we are to be baptized into His name for salvation (Acts 4:10-12); not in titles.
As stated though, scripture CAN'T be in conflict with itself, so, Matthew 28:19 CAN'T be wrong, superceded by Acts 2:38. They must somehow be in harmony which is why I brought this up. In both cases, it's really saying by the authority of, and looking at it like that, they both make sense and harmonize as they should. Another point to consider is found below in a response of mine on this topic posted elsewhere on another forum for your consideration.
REPOST
"To my previous notes suggesting that we today, may be placing too much emphasis on the baptizer and what he says or doesn't say as relates to the process of baptism at the actual time of immersion, as opposed to the emphasis being on the one about to be baptized, and many being concerned that a baptism may be invalid if the baptizer doesn't say the appropriate words, I'll provide some other scriptural information and observation below for further consideration.
Although we today make reference to Matthew 28:19 and/or Acts 2:38 as guides to follow, one saying something different than the other causing confusion and controversy, with either one or the other typically used at the time of baptism in somewhat of a ceremonial process being verbalized by the one performing the baptism, you'll find no record in the scriptures of any such thing occurring at the time of baptism or just prior to the actual act of the baptizer performing the immersion (baptism) of the new believer. Nowhere can be found a baptizer saying "I baptize you in the name of the Father and of the Son and of the Holy Ghost", or, "I baptize you in the name of Jesus Christ" or anything similar or anything at all for that matter. What you do find is simply that the person or persons were simply immediately baptized.
Also, in only two instances recorded, you'll find that anything at all was said or required to be said relative to either of these 2 scriptures, and in both instances, it's the one being baptized that either says something or is told to say something. Those 2 instances are found in Acts 8 and Acts 22.
In Acts 8:34-38 below KJV, you'll see in verse 37 that upon believing, the Ethiopian eunuch confirmed his belief verbally in the presence of only Philip, by stating he believed "that Jesus Christ is the Son of God". Immediately after that, verse 38 days he was baptized, immersed in water. Nothing else was recorded as being said by the baptizer, Philip, prior to the baptism. Note too this is also consistent with the requirement of Romans 10:9-10, that belief and confession are a necessary part of the salvation process.
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34And the eunuch answered Philip, and said, I pray thee, of whom speaketh the prophet this? of himself, or of some other man?
35Then Philip opened his mouth, and began at the same scripture, and preached unto him Jesus.
36And as they went on
their way, they came unto a certain water: and the eunuch said, See,
here is water; what doth hinder me to be baptized?
37And Philip said, If thou believest with all thine heart, thou mayest. And he answered and said, I believe that Jesus Christ is the Son of God.
38And he commanded the chariot to stand still: and they went down both into the water, both Philip and the eunuch; and he baptized him."
Romans 10:9-10
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9That if thou shalt confess with thy mouth the Lord Jesus, and shalt believe in thine heart that God hath raised him from the dead, thou shalt be saved.
10For with the heart man believeth unto righteousness; and with the mouth confession is made unto salvation."
In Acts 22:6-16 below KJV, Saul (Paul) recounts his conversion which is initially recorded in Acts 9:3-19. And you'll notice what is said in verse 16 of this scripture. Upon Paul's belief, Paul was told to "be baptized, and wash away thy sins, calling on the name of the Lord". ("of Him" 'autou', in the original Greek per the interlinear.)
"16And now why tarriest thou? arise, and be baptized, and wash away thy sins, calling on the name of the Lord."
So, it raises questions in my mind, such as, was anything more said by the baptizers of the bible unlike today? Was or is anything really necessary to be said by baptizers today? Does whatever is said 'incorrectly' or thought to be incorrect by baptizers today have effect on the validity of the baptism? Speak where the Bible speaks and remain silent where it is silent. Is this applicable here as well? Were the baptizers silent? Are we as baptizers silent today? Should we be simply immersing those who have heard the gospel, believed it, confessed their belief, and repented? It's a good question. Were the 3000 that believed as noted in Acts 2:37-42, immediately baptized by who knows how many baptizers, or was some sort of ceremonial words spoken before each as they were baptized?"