"Is the Spirit of the Son distinct from God the Holy Ghost?"
Isaiah 9:6 provides affirmation that the Son is indeed the Father:
For 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.
Even the Son's name "shall be called" (future from the days of the prophet) "The mighty God", The everlasting Father, The Prince of Peace. If there is a distinction between the Son and God, the Holy Ghost, what exactly is being distinguished, and what is not?
So let's go right to Genesis chapter 1 (I'm using the KJV):
In the beginning God created the heaven and the earth. And the earth was without form, and void; and darkness was upon the face of the deep. And the Spirit of God moved upon the face of the waters.
And God said, Let there be light: and there was light.
Verse 1: God "created" ; Verse 2: The Spirit of God "moved"; Verse 3: And God "said". Who could possibly contend that God was not performing an OPERATION THAT ONLY GOD COULD PERFORM in each of the first three verses at that point in time--the very beginning of time. Yet, God Almighty performed three distinctly different operations through three distinctly different operative capacities--Almighty God, The Spirit of God, and the Word of God.
Who created God Almighty? Who gave life to the Spirit of God that "moved without any help? Which word twiddler taught God how to speak words? Who fashioned the eyes of the All-Seing God?
There is only ONE Almighty God (being a single plurality), one spirit of God, and one WORD of God. Moreover, throughout Gen. 1, God saw, called, made, and blessed differently on different occasions. Therefore, we are told to understand that God is a Spirit, and we are to worship Him in spirit and in truth. The speaking part of God is also spirit (no duh) and speaks truth (the words that I speak, they are spirit, and they are life),
So the operative capacity of the Father pursuant to His ability to generate offspring is indeed established through His relationship with that offspring--all being living spirits, including Adam--in the image of, and after the likeness of the complex plural Spirit God. Adam was called the son of God in Luke 3:38:
Which was the son of Enos, which was the son of Seth, which was the son of Adam, which was the son of God. (My emphasis)
Adam was indeed "created" as a complex plural spirit as we see in Gen. 5:1-2:
This is the book of the generations of Adam. In the day that God created man, in the likeness of God made he him; Male and female created he them; and blessed them, and called their name Adam, in the day when they were created.
Before Adam was "formed" of the dust of the ground as a single flesh and blood man, he was "created" as that complex plural spirit in God's image and after His likeness. He was first "created" as being both male and female in one spirit--referred to by "their name" which was called "Adam"--not Adam and Eve. Only after that was man "formed" of the dust of the ground. And EVEN LATER THAN THAT was woman taken out of man and "formed" of Adams flesh and bone, as we see affirmed in 1 Timothy 2:13:
For Adam was first formed, then Eve. (My emphasis)
Whenever God is acting within the operative capacity of the Father, He stays in His own lane--being what a Father should be--doing what a Father should do--loving, protecting, and instructing His Son . Likewise, whenever God is acting in the operative capacity of the Son, He stays in His own lane--being what a Son should be--doing what a Son should do--loving and obeying his Father.
Accordingly, when Jesus said, I and my father are one, He was not being disrespectful of His Father. He was stating a fact. Yet, Jesus, in the operative capacity as the Son of God/Son of man, said in Mat. 24:36-37:
But of that day and hour knoweth no man, no, not the angels of heaven, but my Father only. But as the days of Noe were, so shall also the coming of the Son of man be.
Clearly, the Son stayed in His own lane, not encroaching upon the operative capacity of the Father. Moreover, the Spirit/Father (not flesh and blood) shows His Righteousness in John 5:22:
For the Father judgeth no man, but hath committed all judgment unto the Son:
And explaining the operative capacity is directly related to being the Son of man, John 5:27 goes on to explain:
And hath given him authority to execute judgment also, because he is the Son of man. (My emphasis)
Precept upon precept, the Father stayed in His own lane by committing all judgment of mankind to the Son of man, tempted in all manners as we men have been tempted, yet without sin. In the operative capacity of the Spirit Father, God places great value upon righteous judgment.
Accordingly, to narrowly use the term, person, as a distinctly different and separate individual being seems impractical as a means of describing the ONE TRUE GOD. Yet, in all your studies, pay careful attention as to which, if any, operative capacity of God is being used.