Does Luke 1:35 and Matthew 1:20 prove the "genesis" of the Son of God as Socinian Unitarians and Modalists suppose? Was Christ "begotten" in the womb of Mary? What is the biblical definition of "Son of God"?

While it is true that the Holy Spirit miraculously caused the conception in the womb of Mary, it's important to recognize that both Luke 1:35 and Matthew 1:20 do not specifically say that the Holy Spirit "fathered" or "begat" Christ directly. I assume all of us would agree that it was Mary herself who "begat" or conceived Christ. Begotten can simply mean give birth and she was the one who gave birth to the Messiah.

It is likely that the ancient Jews viewed "Son of God" as a messianic title wherein the Messiah is the beloved son and representative of God. This of course is true about Jesus (even in His humanity) but He fulfills the title "Son of God" in the fullest sense: He's the Son of God by nature and is therefore equal with God (John 5:18). Son of God doesn't have exclusively one definition. It is foolish to suggest that it does. Adam was the son of God in a different sense than Solomon, David or Israel. "'Son of the Most High" is also mentioned in Luke 1:32 in a messianic sense since His kingdom will have no end (v.33). Is "Son of God" in Luke 1:35 totally different from v.32? Luke 1:35 is also likely pointing us to Isaiah where a child is born and a 'son' is given and the virgin conceiving and bearing a son (7:14). Again highlighting the idea that Luke is using this title in a messianic sense.

Additionally, Oneness Pentecostals sometimes like to suggest Luke 1:35 means the Holy Spirit Himself became flesh as the Son of God. This is a big stretch of the language of the verse. They are reading their own theology into it. The verse simply doesn't mean that.