You ask, "How do Trinitarians explain why the title "Father" is given to a person in the Trinity other than "The Holy Spirit"?
You see Matthew 1:18 & 20 as stating the Holy Spirit to be the Father of Jesus Christ. That's not how Trinitarians understand those verses, but it's intriguing that you do, and I just want to take it from there.
The Bible calls the son that was to be born "the Everlasting Father" in the prophecy of Isaiah 9:6. So, there you have the Son given the title of "Father".
The Bible calls the other person of the Godhead "Father" as well! One of dozens of examples is Matthew 3:16-17 where the Holy Spirit appeared as a dove over Jesus' head, and then came "a voice from heaven, saying, 'This is my beloved Son, in whom I am well pleased" - the Father in heaven (not the Holy Spirit on earth at that point.)
This means that the Father and the Son are both called "Father", plus you think the Holy Spirit is also called "Father". On the basis of your own belief (that the Holy Spirit is called "Father"), you now have to face the point that the other two persons are equally called "Father". This is a point of logic, based on your own comments. Official trinitarian explanations cannot be gone into until you either agree or disagree with the logic of this.
You also hope there will be "an explanation of whether this event was effected by God's impersonal spirit (power), or by the third Person." However, it is illogical to expect a trinitarian explanation of something that trinitarians do not believe - that God has impersonal spirit power, for that is what groups like the Jehovah's Witnesses believe, and they are anti-trinitarian.
This means I can hardly get going with your question because your rationale is at odds with trinitarian thinking. But I've done the best I can, based on your own comments. Please do not start asking me more questions in comment boxes below, for - really - you need to ask an entirely new question, for you are asking on the basis of what you think, and not what trinitarians think. They call the Father "Father" because the Bible does. That's the simple answer to your main question.