A basic Definition:
“Within the one Being that is God, there exists eternally three coequal and coeternal Persons, namely, the Father, the Son, and the Holy Spirit”
“Being and Person are different thing: Being is what makes something what it is, Person is what makes someone who they are.”
The most common misunderstanding:
“The Trinity is not saying that there are three Beings who are one Being, nor three Persons who are one Person. One Being, eternal and unlimited, shared by three divine Persons, Father, Son, and Holy Spirit.”
- Dr. James White
Begotten Son of God defined
The Meaning of “Only Begotten”:
This is the proclamation formulated by the Nicene Council in the following way:
“Begotten not made. Of one essence with the Father also called the Word or Logos of God, before His incarnation. There never was a time when there was no Son of God. Always “exists” as uncreated, eternal, and divine.”
[R. Alan Cole, “Begotten,” The Zondervan Pictorial Encyclopedia of the Bible (Grand Rapids: Zondervan, 1976), p. 510.]
6For unto us a child is born, unto us a son is given: and the government shall be upon his shoulder: and his name shall be called Wonderful, Counsellor, The mighty God, The everlasting Father, The Prince of Peace. - Isaiah 9 (King James Version)
14And the Word was made flesh, and dwelt among us, (and we beheld his glory, the glory as of the only begotten of the Father,) full of grace and truth.- John
6Who, being in the form of God, thought it not robbery to be equal with God:
7But made himself of no reputation, and took upon him the form of a servant, and was made in the likeness of men: - Philippians 2:6-7 (King James Version)
35And the angel answered and said unto her, The Holy Ghost shall come upon thee, and the power of the Highest shall overshadow thee: therefore also that holy thing which shall be born of thee shall be called the Son of God. - Luke 1 (King James Version)
The Incarnation refers to the moment Jesus was conceived within Mary’s womb.
For a matter of fact, textually, it was Mary who beget Jesus.
What about the Father?
After Adam and Eve ate the forbidden fruit, according to Genesis, "they heard the sound of the Lord God walking in the garden at the time of the evening breeze, and the man and his wife hid themselves from the presence of the LORD GOD among the trees of the garden." Hiding may sound like a naive strategy to deploy against the omniscient God we know today, but apparently he wasn't omniscient back then. For "the LORD GOD called to the man, and said to him, "Where are you?" - [Robert Wright's 'The Evolution of God']
So there's a difference between labeling (Dr. James White's definition) and content (scriptures).