According to Trinitarians, is Jesus the begotten Son of God the Father, or the begotten Son of God?
The short answer is that Jesus is the begotten Son of God the Father!
Although Trinitarians will haggle to some degree over definitions in place here. I would like to give a response based on the Catholic viewpoint. Other denominations, I am sure will have a similar understanding also.
Before going on it would good to be be aware that both translations and interpretation will vary even within a Trinitarian outlook.
Also, to understand how Catholics understand the phrase begotten of the Father, one has to be able to understand the Trinitarian terms such as the Spiration of the Holy Spirit and Perichoresis
Perichoresis (from Greek: περιχώρησις perikhōrēsis, "rotation")1 is a term referring to the relationship of the three persons of the triune God (Father, Son, and Holy Spirit) to one another. Circumincession is a Latin-derived term for the same concept. It was first used as a term in Christian theology, by the Church Fathers. The noun first appears in the writings of Maximus Confessor (d. 662) but the related verb perichoreo is found earlier in Gregory of Nazianzus (d. 389/90). Gregory used it to describe the relationship between the divine and human natures of Christ as did John of Damascus (d. 749), who also extended it to the "interpenetration" of the three persons of the Trinity, and it became a technical term for the latter. It has been given recent currency by such contemporary writers as Jürgen Moltmann, Miroslav Volf, John Zizioulas, Richard Rohr, and others.
Modern authors extend the original usage as an analogy to cover other interpersonal relationships. The term "co(-)inherence" is sometimes used as a synonym.
Since humans are made in the image of God, a Christian understanding of an adequate anthropology of humans' social relations is informed by the divine attributes, what can be known of God's activity and God's presence in human affairs. Theologians of the Communio school such as Hans Urs von Balthasar, Henri de Lubac, and Joseph Ratzinger (later Pope Benedict XVI) locate the reciprocal dynamism between God and God's creatures in the liturgical action of sacrament, celebrating the sacred mysteries in Eucharistic communion, in a hermeneutic of continuity and apostolic unity.
Gothic triskele window element
It is a defined dogma of the Catholic Church that the Holy Spirit proceeds from the Father and from the Son as from a single principle through a single spiration. You will find this in Lateran IV (D 428), Lyons II (D 460, 463), and Florence (D 691, 703, 704). That the Holy Spirit is not generated and so not a son is affirmed by the Athanasian Creed (D 39), Toledo XI (D 277), and Lateran IV (D 428).
In the New Testament, the Holy Spirit is said to be the Spirit of the Father and the Spirit of the Son. Therefore he proceeds from both the Father and the Son. He is called the Spirit of the Father in Matt. 3:16; 10:20; Rom. 8:9; 1 Cor 2:12. He is called the Spirit of the Son (or of the Lord or of Jesus) in Acts 5:9; 16:7; 2 Cor. 3:17f; Gal. 4:6; Phil 1:19; Rom 8:9-11. These texts speak about a divine Person, not a mere created gift.
Tim Staples has a good take on this issue in his article: Explaining the Trinity. St. Thomas Aquinas explains, and Scripture reveals, the Son is uniquely “begotten” of the Father!
Processions and Relations in God
In Catholic theology, we understand the persons of the Blessed Trinity subsisting within the inner life of God to be truly distinct relationally, but not as a matter of essence, or nature. Each of the three persons in the godhead possesses the same eternal and infinite divine nature; thus, they are the one, true God in essence or nature, not “three Gods.” Yet, they are truly distinct in their relations to each other.
In order to understand the concept of person in God, we have to understand its foundation in the processions and relations within the inner life of God. And the Council of Florence, AD 1338-1445, can help us in this regard.
The Council’s definitions concerning the Trinity are really as easy as one, two, three… four. It taught there is one nature in God, and that there are two processions, three persons, and four relations that constitute the Blessed Trinity. The Son “proceeds” from the Father, and the Holy Spirit “proceeds from the Father and the Son.” These are the two processions in God. And these are foundational to the four relations that constitute the three persons in God. These are those four eternal relations in God:
The Father actively and eternally generates the Son, constituting the person of God, the Father.
The Son is passively generated of the Father, which constitutes the person of the Son.
The Father and the Son actively spirate the Holy Spirit in the one relation within the inner life of God that does not constitute a person. It does not do so because the Father and Son are already constituted as persons in relation to each other in the first two relations. This is why CCC 240 teaches, “[The Second Person of the Blessed Trinity] is Son only in relation to his Father.”
The Holy Spirit is passively spirated of the Father and the Son, constituting the person of the Holy Spirit.
We should take note of the distinction between the “generative” procession that constitutes the Son, and the “spirative” procession that constitutes the Holy Spirit. As St. Thomas Aquinas explains, and Scripture reveals, the Son is uniquely “begotten” of the Father (cf. John 3:16; 1:18). He is also said to proceed from the Father as “the Word” in John 1:1. This “generative” procession is one of “begetting,” but not in the same way a dog “begets” a dog, or a human being “begets” a human being. This is an intellectual “begetting,” and fittingly so, as a “word” proceeds from the knower while, at the same time remaining in the knower. Thus, this procession or begetting of the Son occurs within the inner life of God. There are not “two beings” involved; rather, two persons relationally distinct, while ever-remaining one in being.
The Holy Spirit proceeds from the Father and the Son, but not in a generative sense; rather, in a spiration. “Spiration” comes from the Latin word for “spirit” or “breath.” Jesus “breathed on them, and said to them, “Receive the Holy Spirit…” (John 20:22). Scripture reveals the Holy Spirit as pertaining to “God’s love [that] has been poured into our hearts” in Romans 5:5, and as flowing out of and identified with the reciprocating love of the Father for the Son and the Son for the Father (John 15:26; Rev. 22:1-2). Thus, the Holy Spirit’s procession is not intellectual and generative, but has its origin in God’s will and in the ultimate act of the will, which is love.
As an infinite act of love between the Father and Son, this “act” is so perfect and infinite that “it” becomes (not in time, of course, but eternally) a “He” in the third person of the Blessed Trinity. This revelation of God’s love personified is the foundation from which Scripture could reveal to us that “God is love” (I John 4:8).
God is not revealed to “be” love in any other religion in the world other than Christianity because in order for there to be love, there must be a beloved. From all eternity, the Father, Son, and Holy Spirit have poured themselves out into each other in an infinite act of love, which we, as Christians, are called to experience through faith and the sacraments by which we are lifted up into that very love of God itself (Romans 5:1-5).
It is the love of God that binds us, heals us, and makes us children of God (I John 4:7; Matt. 5:44-45). Thus, how fitting it is that the Holy Spirit is depicted in Revelation 22:1-2, as a river of life flowing out from the Father and the Son and bringing life to all by way of bringing life to the very “tree of life” that is the source of eternal life in the the Book of Revelation (Rev. 22:19).