Well, the rendering of "born again" in 3:5 does seem to be exclusive to the old Vulgate and the translations derived from it. Nevertheless, the sense is the same.
As for what you are trying to get at with the distinction between those "born of the Spirit" and "begotten of the Spirit," I can't make any sense of it. It certainly isn't in 2 Cor. 4:18.
Men are "born of water and the Holy Spirit" in baptism, not at birth. The first birth is that of the corrupted seed of Adam, the second in the laver of regeneration, the washing of the water with the word, in which we are washed, sanctified, and justified (Cf. Titus 3:5; Eph. 5:26; 1 Cor. 5:11). This is the birth without which one can neither see nor enter the kingdom of God.
1 Erat autem homo ex pharisaeis, Nicodemus nomine, princeps Iudaeorum;
2 hic venit ad eum nocte et dixit ei: “ Rabbi, scimus quia a Deo venisti magister; nemo enim potest haec signa facere, quae tu facis, nisi fuerit Deus cum eo ”.
3 Respondit Iesus et dixit ei: “ Amen, amen dico tibi: Nisi quis natus fuerit desuper, non potest videre regnum Dei ”.
4 Dicit ad eum Nicodemus: “ Quomodo potest homo nasci, cum senex sit? Numquid potest in ventrem matris suae iterato introire et nasci? ”.
5 Respondit Iesus: “ Amen, amen dico tibi: Nisi quis natus fuerit ex aqua et Spiritu, non potest introire in regnum Dei.
6 Quod natum est ex carne, caro est; et, quod natum est ex Spiritu, spiritus est.
7 Non mireris quia dixi tibi: Oportet vos nasci denuo.
8 Spiritus, ubi vult, spirat, et vocem eius audis, sed non scis unde veniat et quo vadat; sic est omnis, qui natus est ex Spiritu ”.
John 3:16, [For God so loved the world that he gave his only Begotten Son].....even Jesus was not "born again" until his resurrection.
2 Corinthians 4:18 says "what is unseen is eternal"... like those born of the spirit.