Is Mary, "Mother of God," the mother of the Son or of the whole Trinity?

I find the Catholic title "Mother of God" for the Virgin Mary confusing

It shouldn't be. Let's rephrase 'Mother of God':

Of Whom was Mary the mother?

If you must withold 'God,' then you withold it of Christ who is very God (in addition to be truly human, or vice versa). And would be a heretic: a Nestorian, to be precise.

If you say 'Jesus, in His humanity' you withold 'God' from Christ, which is impossible: He being at all times absolutely God and absolutely man, and never only either, since His incarnation—since the Hypostatic Union.

Galatians 4:4

But when the fulness of the time was come, God sent his Son, made of a woman...

This Son was at all times 'God' and so Mary, being His mother, is indeed the Mother of God. It's really very simple.

Similarly,

Luke 1:43

And whence is this to me, that the mother of my Lord should come to me?

Now if one doubts this Lord is Christ, or that He is "Lord..and..God" (John 20:28) Then He has a faulty Christology. As well as a misunderstanding of the doctrine of the Trinity. The doctrine of Trinity is such that each Person of the Trinity is truly God, individually, but not separate Gods. Nonetheless, they are each God because they are of the one substance, God. each Person of the Trinity is not the Trinity; but each Person of the Trinity is God, fully, truly.

"This problem doesn't exist when Mary is only mother of Jesus, because then it could be merely a way of speaking to say that Mary was the one through whom the Word became flesh"

Not if "the Word was God" (John 1:1) is in your Bible.

One needs to make the "the Word" who was and is "God" different from the one "born of [Mary]" (Gal 4:4) in order for her not to be the Mother of God.

According to this syllogism, she is the Mother of God, not just the Mother of the Son of God:

1. The Son of God is God. (1 Jn. 5:20)
2. Mary is the mother of the Son of God. (Mat. 1:18)
3. Ergo, Mary is the Mother of God.

Mary does not precede God, since nothing precedes Him; He is the Alpha, the First. Christ's humanity was created in Mary's womb; God is uncreated. Yet, because Christ's humanity is united to His divinity in the hypostatic union, Mary is truly the mother of both His humanity and divinity.

Just because God creates a newly conceived person's soul out of nothing, does this mean this person's biological mother is only the mother of "half" the person, viz., of his body only? Analogously with the Mother of God and Christ: Just because Mary didn't create the (uncreated) Divinity of God in her womb, does that make her only the mother of His humanity (and just of his human body, per the aforementioned reason)?

Addressing the question "Whether the Blessed Virgin should be called the Mother of God?," St. Thomas Aquinas gives the following objection:

Christ is called God in respect of His Divine Nature. But the Divine Nature did not first originate from the Virgin. Therefore the Blessed Virgin should not be called the Mother of God.

to which he replies with a quote from St. Cyril, similar to the questions I asked above:

This was an argument of Nestorius. But Cyril, in a letter against Nestorius [*Cf. Acta Conc. Ephes., p. 1, cap. ii], answers it thus: "Just as when a man's soul is born with its body, they are considered as one being: and if anyone wish to say that the mother of the flesh is not the mother of the soul, he says too much. Something like this may be perceived in the generation of Christ. For the Word of God was born of the substance of God the Father: but because He took flesh, we must of necessity confess that in the flesh He was born of a woman." Consequently we must say that the Blessed Virgin is called the Mother of God, not as though she were the Mother of the Godhead [mater divinitatis or "Mother/Creator of Divinity"], but because she is the mother, according to His human nature, of the Person who has both the divine and the human nature.