I point to the nicene creed (to make things simpler hopefully to start with) I point to the version which speaks of the holy spirit proceeding from the father (alone):
What is the difference in the nature of how the Son is generated versus how the Holy Spirit is generated? Obviously there needs to be a difference otherwise there would be two Sons which is not in accordance with Church teaching…
…perhaps put in another way, what is the logical precedence in the trinity (Here I mean, that the three persons of the trinity are co-equal and co-eternal in actuality but the father has a logical precedence over the son in the sense that son depends upon being generated by the father. Where does the holy spirit fit into this)?
God created man and woman in God’s image and likeness. Man breathes and speaks.
When God speaks and breathes from His Essence Existing eternally (metaphorically speaking here)
The distinction is revealed in the Voice that is heard from God the Father, what proceeds within and from that voice is God the Son who is the Word of God proceeding, generating revelation of God’s presence. What proceeds from God’s Voice, distinctly from God’s Word is God’s Breath =God the Holy Spirit who proceeds from both the Voice and the Word in presence to make known the revelation of God from God’s presence.
Does this explain the Trinity, NO, not even close. It is just a reflection of What God has revealed to our humanity with God’s presence in space and time =human history.
The Trinity is never divided because the voice, the word, the breath are all one God in being. Yet each one is distinct in revelation of the other, the voice is God, the word is God, the breath is God. They are distinct in procession, generation, revelation of God’s presence, never divided, not separated or confused One God in Essence being.
Things to consider; The Son eternally proceeds from the Father is a mystery that relates to God’s Essence being Existence itself which eternity obeys and space and time cannot comprehend.
Thus God’s Essence is not revealed in space and time, but God’s presence is revealed in space and time with God’s presence in the Father (voice), the Son (Word) and the Holy Spirit (breathe), when all three are distinct in presence , all three distinct persons is God’s Essence being, for the presence of God proceeds in Trinity to our humanity. While God’s presence veils God’s Essence. When no one can see God and live. But scripture reveals only the Son has seen the Father because the Son comes from the Father and the Father and the Son is one, this is spoken of an eternal relationship with the Holy Spirit proceeding from both in Trinity, Life breathes life eternally to give life to our humanity, when we profess God from God, Light from Light.
Eternal procession, presence, substance=Essence= divinity, eternal begetting, eternally begotten, sending and proceeding, generating are terms that point to God’s presence existing in eternity being made present in space and time. God has revealed this throughout human history recorded, God reveals His presence in space and time in Trinity, the voice, word and breath of God reveal persons living and being God, the Father, God the Son and God the Holy Spirit.
Peace be with you