You are right to notice the importance of this difference, especially as it is a theological commonplace. There is no underlying textual differences in the Greek. It is a translation choice that has emerged from another theological tradition. If you would like to see a sound exposition I recommend John Owen, who also provided a solid refutation of the false interpretation.
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John Owen comments:
I. The Lord Jesus Christ is truly God and man in one person; and this is fully manifested in these words. For,
1. There is supposed in them his
pre-existence in another nature than that which he is said here to assume. He w
as before, he
subsisted before, or he could not have taken on him what he had not. This was his divine nature; as the like is intimated where he is said to be “made flesh,”
John 1:14; to be “made of a woman,”
Galatians 4:4; to be “manifested in the flesh,”
1 Timothy 3:16; to “take on him the form of a servant,”
Philippians 2:8-9; as here, “he took the seed of Abraham.” He
was before he did so; that is, as the Son, the Word of God, the Son of God, as in the places mentioned, eternally pre-existing unto this his incarnation: for the subject of this proposition, “He took on him,” etc., denotes a person pre-existing unto the act of taking here ascribed unto him; which was no other than the Son of God.
2. He assumed, he took to himself, another nature, “of the seed of Abraham,” according unto the promise. So, continuing what he was, he became what he was not. For,
3. He took this to be his
own nature. He so took it as himself to become truly “the seed of Abraham,” to whom and concerning whom the promise was given,
Galatians 3:16; and was himself made “of the seed of David according to the flesh,”
Romans 1:3; and “as concerning the flesh came of the fathers,”
Romans 9:5; and so was “the son of David, the son of Abraham,”
Matthew 1:1. And this could no otherwise be done but,
4. By taking that nature into
personal subsistence with himself, in the hypostasis of the Son of God. The nature he assumed could no otherwise become his. For if he had by any ways or means taken the person of a man to be united unto him, in the strictest union that two persons are capable of, a divine and a human, the nature had still been the nature of that other person, and not his own.
5. But he took it to be
his own nature; which it could no ways be but by personal union, causing it to subsist in his own person. And he is therefore a true and perfect man: for no more is required to make a complete and perfect man but the entire nature of man subsisting; and this is in Christ as a man, the human nature having a subsistence communicated unto it by the Son of God. And therefore,
6. This is done without a
multiplication of persons in him; for the human nature can have no personality of its own, because it was taken to be the nature of another person who was pre-existent unto it, and by assuming of it prevented its proper personality. Neither,
7. Did hence any mixture or confusion of natures ensue, or of the essential properties of them; for he took the seed of Abraham to be his human nature, which if mixed with the divine it could not be. And this he hath done,
8. Inseparably and for ever. Which things are handled at large elsewhere.
