Fact is whatever word Jesus used in Hebrew Nicodemus understood Jesus was saying “again” and wondered how that could be. So “born again” is correct both by John’s Greek translation of Jesus’ words and the context of how Nicodemus responded.
Fact you say? Whoever wrote John's Gospel wasn't even there, nor does he claim he was. You are just assuming that Jesus was whispering in "John's " ear as was he was writing.
The Gospel of John is, as we have it, is heavily interpolated (i.e. edited) by someone else and does not always appear in order or even a logical sequence. Even as early as circa AD 250, Dionysius, bishop of Alexandria, argued that —based on the vocabulary, thought and style—the Gospel of John had two authors.
The fact is that the Nicodemus' reply makes no sense. Appeals to a translational error John made, as a way of rationalizing this problem away, makes even less sense when one remembers that "John" was writing under "inspiration" and therefore free or error (another presumption), and that therefore he could not have mistranslated.
Jesus' reply in John 3:5 only confirms that he meant from above (i.e. from heaven) and not again: "unless one born of the water and the spirit he cannot enter into he kingdom of God" which is saying somewhat different from John 3:3, but in either case Jesus never says one must be born again.
The other problem, besides Nic's off the wall response, is that the term "kingdom of God" in Judaism means simply Israel! Christians changed this to mean something else. So, as a Jew to a Jew, assuming this conversation ever took place, Jesus was telling Nicodemus that one must be born from above, that is from the water the spirit if he is to enter Israel! Clearly none of this makes any sense if we assume he was talking to Nicodemus in any language.