What did the early church fathers (pre-5th c.) teach with regards to baptismal regeneration?

Justin Martyr wrote a chapter (LXI) on Christian baptism in his First Apology, written sometime in the 2nd century:

I will also relate the manner in which we dedicated ourselves to God when we had been made new through Christ; lest, if we omit this, we seem to be unfair in the explanation we are making. As many as are persuaded and believe that what we teach and say is true, and undertake to be able to live accordingly, are instructed to pray and to entreat God with fasting, for the remission of their sins that are past, we praying and fasting with them. Then they are brought by us where there is water, and are regenerated in the same manner in which we were ourselves regenerated. For, in the name of God, the Father and Lord of the universe, and of our Saviour Jesus Christ, and of the Holy Spirit, they then receive the washing with water. For Christ also said, Except ye be born again, ye shall not enter into the kingdom of heaven [John 3:5]. Now, that it is impossible for those who have once been born to enter into their mothers’ wombs, is manifest to all. And how those who have sinned and repent shall escape their sins, is declared by Isaiah the prophet, as I wrote above, he thus speaks: Wash you, make you clean; put away the evil of your doings from your souls; learn to do well; judge the fatherless, and plead for the widow: and come and let us reason together, saith the Lord. And though your sins be as scarlet, I will make them white like wool; and though they be as crimson, I will make them white as snow. But if ye refuse and rebel, the sword shall devour you: for the mouth of the Lord hath spoken it [Isaiah 1:16-20].

And for this [rite] we have learned from the apostles this reason. Since at our birth we were born without our own knowledge or choice, by our parents coming together, and were brought up in bad habits and wicked training; in order that we may not remain the children of necessity and of ignorance, but may become the children of choice and knowledge, and may obtain in the water the remission of sins formerly committed, there is pronounced over him who chooses to be born again, and has repented of his sins, the name of God the Father and Lord of the universe; he who leads to the laver the person that is to be washed calling him by this name alone. For no one can utter the name of the ineffable God; and if any one dare to say that there is a name, he raves with a hopeless madness. And this washing is called illumination, because they who learn these things are illuminated in their understandings. And in the name of Jesus Christ, who was crucified under Pontius Pilate, and in the name of the Holy Ghost, who through the prophets foretold all things about Jesus, he who is illuminated is washed.

Cyprian of Carthage, another Ante-Nicene Father, wrote in the 3rd century of his own baptism:

For as I myself was held in bonds by the innumerable errors of my previous life, from which I did not believe that I could by possibility be delivered, so I was disposed to acquiesce in my clinging vices; and because I despaired of better things, I used to indulge my sins as if they were actually parts of me, and indigenous to me. But after that, by the help of the water of new birth [i.e., baptism], the stain of former years had been washed away, and a light from above, serene and pure, had been infused into my reconciled heart,—after that, by the agency of the Spirit breathed from heaven, a second birth had restored me to a new man;—then, in a wondrous manner, doubtful things at once began to assure themselves to me, hidden things to be revealed, dark things to be enlightened, what before had seemed difficult began to suggest a means of accomplishment, what had been thought impossible, to be capable of being achieved; so that I was enabled to acknowledge that what previously, being born of the flesh, had been living in the practice of sins, was of the earth earthly, but had now begun to be of God, and was animated by the Spirit of holiness (Epistle I, To Donatus).

These are the earliest sources I could find who comment on the nature of baptism. I am sure there are many, many such commentaries among the Post-Nicene Fathers, but rather than search through them I thought I would simply quote what John of Damascus wrote later in his An Exact Exposition of the Orthodox Faith. Written in the early 8th century, Exposition was probably the first summary of Church dogmatic theology and reflects the collected teaching of recognized Church Fathers up to that point. In "Concerning Faith and Baptism", he writes:

We confess one baptism for the remission of sins and for life eternal. For baptism declares the Lord’s death. We are indeed buried with the Lord through baptism [Col 2:12], as saith the divine Apostle. So then, as our Lord died once for all, we also must be baptized once for all, and baptized according to the Word of the Lord, In the Name of the Father, and of the Son, and of the Holy Spirit [Matt 28:19], being taught the confession in Father, Son, and Holy Spirit.

The baptism then into Christ means that believers are baptized into Him. We could not believe in Christ if we were not taught confession in Father, Son, and Holy Spirit4. For Christ is the Son of the Living God5, Whom the Father anointed with the Holy Spirit: in the words of the divine David, Therefore God, thy God, hath anointed thee with the oil of gladness above thy fellows [Ps 44:7 LXX]. And Isaiah also speaking in the person of the Lord says, The Spirit of the Lord is upon me because He hath anointed me [61:1]. Christ, however, taught His own disciples the invocation and said, Baptizing them in the Name of the Father, and of the Son, and of the Holy Spirit. For since Christ made us for incorruption, and we transgressed His saving command. He condemned us to the corruption of death in order that that which is evil should not be immortal, and when in His compassion He stooped to His servants and became like us, He redeemed us from corruption through His own passion. He caused the fountain of remission to well forth for us out of His holy and immaculate side [Jn 19:34], water for our regeneration, and the washing away of sin and corruption; and blood to drink as the hostage of life eternal. And He laid on us the command to be born again of water and of the Spirit [Jn 3:5], through prayer and invocation, the Holy Spirit drawing nigh unto the water. For since man’s nature is twofold, consisting of soul and body, He bestowed on us a twofold purification, of water and of the Spirit: the Spirit renewing that part in us which is after His image and likeness, and the water by the grace of the Spirit cleansing the body from sin and delivering it from corruption, the water indeed expressing the image of death, but the Spirit affording the earnest of life.

For from the beginning the Spirit of God moved upon the face of the waters [Gn 1:2], and anew the Scripture witnesseth that water has the power of purification [Lev 15:10]. In the time of Noah God washed away the sin of the world by water [Gn 16:17]. By water every impure person is purified, according to the law, even the very garments being washed with water. Elijah shewed forth the grace of the Spirit mingled with the water when he burned the holocaust by pouring on water [3 Kingdoms 18:32 LXX]. And almost everything is purified by water according to the law: for the things of sight are symbols of the things of thought. The regeneration, however, takes place in the spirit: for faith has the power of making us sons (of God), creatures as we are, by the Spirit, and of leading us into our original blessedness.

The remission of sins, therefore, is granted alike to all through baptism: but the grace of the Spirit is proportional to the faith and previous purification. Now, indeed, we receive the firstfruits of the Holy Spirit through baptism, and the second birth is for us the beginning and seal and security and illumination of another life