We have lost something on the subject of Baptism due to the early apostasy of the church by which it started to elevate the external forms of religion above its internal truth. Worrying about which words should be used, and trying to argue them from scriptures, is much like the woman asking Jesus on what mountain should people worship God. (John 4:20). In reply, Jesus said:
But the hour is coming, and is now here, when the true worshipers will worship the Father in spirit and truth, for the Father is seeking such people to worship him. God is spirit, and those who worship him must worship in spirit and truth.” (John 4:23-24, ESV)
(Note: I realize the question is not worrying about the words but is merely a question. It is the potential undue nervousness in choosing a supposed 'correct answer' that is not scriptural.)
A better concept about Baptism is reflected on Paul’s own:
For Christ did not send me to baptize but to preach the gospel, and not with words of eloquent wisdom, lest the cross of Christ be emptied of its power. (1 Corinthians 1:17, ESV)
Charles Hodge has some very apt comments on this verse:
For indicates the connection. ‘I baptized few, for I was not sent to baptize, but to preach.’ The commission was, “Go ye into all the world, and preach the gospel to every creature.” This does not mean that baptism was not included, but it does mean that baptizing was very inferior to preaching. It is subordinated in the very form of the commission, “Go ye therefore, make disciples of all nations, baptizing them,” &c. The main thing was to make disciples; recognizing them as such by baptism was subordinate, though commanded. Baptism was a work which the apostles seem to have generally left to others, Acts 10:48. During the apostolic age, and in the apostolic form of religion, truth stood immeasurably above external rites. The apostasy of the church consisted in making rites more important than truth. The apostle’s manner of speaking of baptism in this connection as subordinate to preaching is, therefore, a wonder to those who are disposed unduly to exalt the sacraments. (Exposition on 1 Corinthians, Charles Hodge, verse 17-31)
The fact is the actuals words that might be spoken during baptism are not prescribed in the Bible, but what baptism means is explained. The same is true of the Lord’s Supper, exact words that are to be said when eating the bread and drinking the wine are not defined in the Bible. The truth of the matter is baptism is principally something one does to say communicate that they personally identify with the death and resurrection of Christ, taking Christ as their redeemer. That they believe and accept Jesus as their Messiah and Lord, in public display and confession -- that is what is established by external baptism. The external rite is about being submerged into water to reflect what has already occurred inwardly. It is not about any specific words to be spoken.
What happens in the inner version of Baptism? Well, a sinner by faith in Christ is united to him. This mystical union punishes his sins in Christ, and Christ's perfection and righteousness is charged and imputed upon the sinner, making him have eternal life, apart from any work. This righteous state brings reconciliation with the Father; therefore the Father gives the Spirit to enjoy forever.
Now in the times of Acts mention is only made of being baptized in the name of Jesus Christ. Of course this is the principle meaning for when one receives Christ as the Messiah, they are baptized into the humanity of Christ and therefore reconciled with God. The description of what it means to be baptized (not the words that must be said) simply expands the reconciliation with God to be reconciliation with the Triune God. Therefore there is no difference between baptism in Mathew 28 and all the other references to Baptism in the Bible where the triune aspect is not focused on, such as Acts 1:5, 2:38, 3:41, 8:12-16, 8:36-38, 9:18, 11:16, 16:15, 16:33, 18:8, 19:3-5, 22:16, Romans 6:3, 1 Corinthians 1:13, 1:14-16, 10:2, 12:13, 15:29 & Galatians 3:27.
With regard to the illogical exegesis of scripture used by the Catholic Encyclopedia one can hardly follow. For it was usual in those times, that upon the baptizing of persons, the Holy Ghost came upon them, and endowed them with extraordinary gifts, peculiar to the days of the gospel. Sometimes people received the Holy Spirit before Baptism also, but it became normal afterwards for this reception to occur during baptism. When Paul found some that had not publically received Christ but had only undergone the Baptism of John before and during the very early part of Christ’s ministry he knew they needed to be baptized as followers of Christ, not of John. This would ensure the receive Christ and the Spirit as all other Christian’s had done. This had nothing to do with the words being spoken. Nobody ever received the Spirit in the Bible by having the word ‘Spirit’ spoken, but by having faith in Christ. The whole notion of attributing power to words is a kind of superstition unknown in scripture.
If I had to take a wild guess I would imagine that, if words were used during Baptism, something to the effect of ‘in the name of Christ’ would have been the most direct and logical, and therefore probable. On the other hand the earliest extra biblical record of actual words spoken under a forming tradition are probably found in the Didache (mid to late first century), which uses a Trinitarian formula.
Conclusion: What words did Peter use? As the exact words are not recorded, I confess I do not know, nor are exact words to be used commanded in any place from scripture. The same is true about words to be spoken during the Lord’s supper.