If you do not believe the Holy Spirit is a person, then what is it? What do Bible verses that reference the Holy Spirit or the Spirit of God mean?

Let me answer by examining the 4 verses that were quoted in the OP Question.

• Genesis 1:1-2

The description of the "Spirit of God ... hovering over the waters" is a poetical image that presents the Spirit of God, a power of God, as though it was distinct from God. Not unlike God's Wisdom, which in Proverbs is even personalized "creature" (Prov 8:22).

• John 14:26

Here it is the Evangelist John who personalizes God's Spirit, calling it "Advocate" (Greek παράκλητος, paraklētos)

• Matthew 12:32

It is necessary to put this verse in context, which is that of Jesus exorcising "a demon-possessed man who was blind and mute" (Matt 12:22-32). The unbelieving Pharisees said “It is only by Beelzebul, the prince of demons, that this fellow drives out demons.” But Jesus replies accusing them: "Anyone who speaks a word against the Son of Man will be forgiven, but anyone who speaks against the Holy Spirit will not be forgiven, either in this age or in the age to come."

The "unforgivable sin" is not the "rejection of Jesus Christ" (and so says with crystal clarity Jesus himself), but the blasphemy against God's Holy Spirit, that is the refusal of recognizing God's presence at work, and the attribution of God's signs even to Beelzebub.

• John 16:7-15

All this pericope appears to refer to the "Advocate", to the Spirit of truth as though it was a person. See above, John 14:26.

To approach an understanding of what is meant by the spirit of God, Holy Spirit, or spirit as mentioned in the Bible is an undertaking worthy of every believer’s best research and thinking. For those of us who are convinced the Bible is given by inspiration of God, a vast amount of comprehension in the scriptures is potentially brought forth. For the skeptic or newcomer, even a basic understanding of this may cause passages that were previously a “closed book” to now be open. Consider the first man, Adam, who God formed, made, and created, as recorded in Isaiah 43:7. He accomplished the amazing task of naming all the animals.

And Adam gave names to all cattle, and to the fowl of the air, and to every beast of the field… Genesis 2:20

In addition, Adam and Eve were given dominion over all the earth. To understand how God accomplished these incredible works in the first man and woman is to begin to understand God’s identity as the Holy Spirit and how He designed the holy spirit He creates in man to operate. Consider the following truths:

• Adam’s body was formed, his soul (which houses the mind) was made, and his spirit (which God can speak to) was created
• God does not speak directly to our minds. Because He is Spirit, he speaks to our spirit (which He creates in us), and our spirit speaks to our mind
• A spirit cannot be seen, heard, etc by one’s five senses. This includes God, the Holy Spirit.
• The knowledge God gives is completely reliable, far exceeding all senses and scientific knowledge
• Man, having free will, has the choice to live by words that come from God or to live by his senses and factual knowledge

Because God had created spirit within them, Adam and Eve were able to talk freely with Him and access divine knowledge. As long as they lived by the knowledge from God they had dominion over all God's creation, and achieved great accomplishments, like naming the animals with names which we use to this day. Four of the five truths of this brief introduction directly apply in the age of grace that we live in today, causing many New Testament passages to open to us. Further knowledge regarding this topic is available in the Internet Archive study: Body, Soul, Spirit - Formed, Made, Created

The Q asked is a simple one that requires little biblical effort to answer. However, the answer will oppose long-held beliefs which run contrary to the testimony of Scripture.

There is no clear, unambiguous case for a divine 'person' the Holy Spirit who is separate from the Father. There is only supposition and extrapolation to arrive at a view that there IS a separate 'person' who is God, but not the Father. Considerable 'reading-in' is required to make a case for a Holy Spirit person. Certainly, some verses may lend themselves to such a conclusion, but only when read in isolation.

One scripture does not stand alone to deduce truth or doctrine, but must be included in a broad range of scripture to get the most authentic understanding.

The disciples had no problem understanding what the Holy Spirit was. They certainly didn’t make more of it than what they learnt from Jesus.

for the Holy Spirit will teach you in that very hour what you ought to say." Luke 12:12

for it will not be you speaking, but the Spirit of your Father speaking through you. Matthew 10:20

We see here the two terms used refer to the same thing. The HS is the power and presence of God. If we choose to make up a person we must ignore these verses to do it.

The HS -

• is never greeted, not sends greetings by/through any NT author.
• has no name. God names everything - all the stars, but not His spirit!
• proceeds from the Father - not God! John 15:26
• doesn't know basic things - in fact is oblivious to essential things Mark 13:32, Luke 10:22 ('no one knows the son except the Father')
• being called 'he' and 'him' is unwarranted in our bibles. Unless it is referring to God or the Father specifically, then 'He' etc would be appropriate. Matt 1:20 Heb 9:8
• when referred to as 'a' gift etc., 'it' or 'which' would be more appropriate. Acts 10:45, Luke 11:20
• needs some solid scriptural support for a 'person', hence 1 John 5:7 has been extended to supply that which God never intended. (even though now corrected in most bibles, some shamelessly or ignorantly still quote this)
• is missing from every vision of the throne whether Jesus is there or not. God is invisible, yet He is 'seen' there with His firstborn human son.
• is never taught by the Apostles to be a 'person', but is made dogma centuries after the church began.

There are many more examples of the HS being expressed as the power and presence of God, and now of Jesus also. Jesus is now expressed AS the spirit (1Cor 3:17) - being the essence of God in human form.

No, there is no need to see the Holy Spirit as distinct from the Father or God. 1 Cor 8:6 pronouncing the truth clearly and providing no reason at all to add another person to this dynamic duo of Father and son.

for us there is but one God, the Father, from whom all things came and for whom we live; and there is but one Lord, Jesus Christ, through whom all things came and through whom we live.

Are we to consider that the 'spirit of' truth, wisdom, Christ, etc are separate entities? Of course not. So why should the Spirit of God be separate from God the Father?

Therefore, since He (Jesus) has been exalted at the right hand of God, and has received the promise of the Holy Spirit from the Father, He has poured out this which you both see and hear. Acts 2:33

So there is no 'person', unless we make one up from poorly considered verses that are persuaded to fit the dogma - while ignoring all those that are quite clearly contrary and explicit in not making the Holy Spirit a person.

None of the verses cited by OP insist on a 'person', unless it is read-in.

John 14:26 for example has 'ho' - this is translated 'which' in most other places so the publishers bias has provided 'he' or 'whom'. This widespread use of personal pronouns confuses correct understanding.

More here

When reading translations of scripture, it is important to remember that the original Greek and Hebrew text didn't use capital letters to personify the "holy spirit" expression. It is the translators, inspired by the Trinity doctrine, that decided to add the capitalization. Similarly, the use of "he", "who", etc. rather than "it", "which", etc. in reference to God's spirit is also a decision of the translators.

For instance, Jewish translations of the Hebrew scriptures do not capitalize "spirit" nor refer to it as "he".

Compare the passage quoted in the question:

But when he, the Spirit of truth, comes, he will guide you into all the truth. He will not speak on his own; he will speak only what he hears, and he will tell you what is yet to come. He will glorify me because it is from me that he will receive what he will make known to you. All that belongs to the Father is mine. That is why I said the Spirit will receive from me what he will make known to you.”

with what it looks like without personification:

But when it, the spirit of truth, comes, it will guide you into all the truth. It will not speak on its own; it will speak only what it hears, and it will tell you what is yet to come. It will glorify me because it is from me that it will receive what it will make known to you. All that belongs to the Father is mine. That is why I said the spirit will receive from me what it will make known to you.”

Both translations are equally correct, but depend on the influence, or lack thereof, of the Trinity doctrine.

Anyone reading the second version would have no reason to even suspect that "spirit" refers to a sentient living being. It doesn't, but the "spirit" in this passage could just as easily be referring to "The Bible" in the way the words are used here.

Read in this way, God's spirit is simply a spiritual force that belongs to God and through which God acts. The expression is treated no differently than "God's finger" or "God's judgement" would be.

• Allowing preconceived ideas to influence one's understanding is known as eisegesis (using scripture to confirm what one knows).
• Looking at scriptures without the influence of preconceived ideas is known as exegesis (using scripture to derive what one knows).

In terms of strengthening one's existing faith, eisegesis can help and exegesis can hurt. But in terms of objective translation, eisegeis is bad and exegesis is good.