Has the term "born again" always been synonymous with baptism with the Christians of the Early Church?

Some Clarification

You question specifically asks if there were any in the Early Church (~2nd to 4th century) who separated "born again" with baptism. Meanwhile, the site you quote claims that this separation took place in the 17th century. Because your question is only concerned with the Early Church, I will not consider any of the possible historical data from the 5th to 17th century.

It is also a bit dangerous to say anyone regards "born again" as being synonymous with baptism. They are two difference terms which have slightly difference meanings--even to those believing in baptismal regeneration. It is similar to how "regeneration" means something slightly different than "born again" despite the fact that most Christians regard them as mostly describing the same event.

Therefore, I take it that when you ask if early Christians all thought that "born again" was synonymous with baptism that you mean to ask if early Christians all thought that "born again" generally describes the event of baptism.

Perhaps a better way to have asked the question would been to ask if there were any early Christians who did not believe in baptismal regeneration. But no worries, I will still answer the question in the terms you've used.

The Answer

The short answer to your question is that there is absolutely no historical data from the Early Church demonstrating that any early Christians understood "born again" as being significantly distinct from baptism. On the contrary, there are a plethora of quotes from the Early Church demonstrating that these Christians understood the two terms to be nearly synonymous. Here are a few.

Theophilus of Antioch in 181 CE

Moreover, those things which were created from the waters were blessed by God, so that this might also be a sign that men would at a future time receive repentance and remission of sins through water and the bath of regeneration -- all who proceed to the truth and are born again and receive a blessing from God. (To Autolycus 2:16)

Irenaeus in 190 CE

"And [Naaman] dipped himself...seven times in the Jordan" [2 Kings 5:14]. It was not for nothing that Naaman of old, when suffering from leprosy, was purified upon his being baptized, but [this served] as an indication to us. For as we are lepers in sin, we are made clean, by means of the sacred water and the invocation of the Lord, from our old transgressions, being spiritually regenerated as new-born babes, even as the Lord has declared: "Except a man be born again through water and the Spirit, he shall not enter into the kingdom of heaven." (Fragment 34)

Recognitions of Clement 221 CE. Notice in this quote that the author mis-remembers John 3:5 leaving out the critical "and of the Spirit."

But you will perhaps say, "What does the baptism of water contribute toward the worship of God?" In the first place, because that which has pleased God is fulfilled. In the second place, because when you are regenerated and born again of water and of God, the frailty of your former birth, which you have through men, is cut off, and so ...you shall be able to attain salvation; but otherwise it is impossible. For thus has the true Prophet [Jesus] testified to us with an oath: "Verily, I say to you, that unless a man is born again of water [sic] he shall not enter into the kingdom of heaven." (Recognitions 6:9)

Cyprian of Carthage 200-258 CE

[When] they receive also the Baptism of the Church...then finally can they be fully sanctified and be the sons of God...since it is written, "Except a man be born again of water and of the Spirit, he cannot enter into the kingdom of God." (Letters 71[72]:1)

Conclusion

There are many more quotes I could fish up for you, but I believe I have given you enough of a starting point. While we could never be historically certain that no early Christians thought "born again" and baptism were significantly distinct, it is indeed very odd that there are no surviving documents demonstrating their existence.