In what sense was mankind created in the image of a triune essence?

Yes, humans were created by God as triune beings. Having created us so, God is then able to interact with us according to all three dimensions of our being. In computer language terms, we implement three interfaces.

The full description of the three clusters of capabilities and realities that define each aspect of our beings is impossible. A few of the are:

• Son. Intellect, knowledge, logic, verbal consciousness, ability to process information and comprehend truth.
• Spirit. Emotion, desire, the need to be comforted and ability to comfort others. The need to receive counsel and ability to counsel others.
• Father. Strength, manipulation of the material world, discipline, habits of action, priorities, perception of time and time management, planning.

This division is based on Jesus’ parable of the soils and other places where the distinct roles of the Trinity are described, such as the Holy Spirit being called the comforter and counselor, Jesus called the Word, and the Father being called one who disciplines his children, as well as being called “the one who is, and was, and is to come”, a reference to time. Also, when asked about when Israel would be restored, Jesus said only the father knows the times.

The New Creation

We are imperfect creatures now. When we are changed at the sounding of the Last Trumpet, we shall model a Triune being perfectly.

The mind of Christ.

“For who has understood the mind of the Lord so as to instruct him?” But we have the mind of Christ. (1 Corinthians 2:16, ESV)

The Indwelling Spirit.

A Glorified body.

So is it with the resurrection of the dead. What is sown is perishable; what is raised is imperishable. It is sown in dishonor; it is raised in glory. It is sown in weakness; it is raised in power. It is sown a natural body; it is raised a spiritual body. If there is a natural body, there is also a spiritual body. Thus it is written, “The first man Adam became a living being”; the last Adam became a life-giving spirit. (1 Corinthians 15:42-45, ESV)

When all this is accomplished, the three aspects of our unitary person will be infused with the actual Trinity. We do not need analogous copies of God’s three persons to make us Triune, as part of the vine we commune with the originals. We are now earthen vessels that contain the essence of God; we shall be glorified vessels. We do not need three vessels to embrace the three persons of the Trinity, only one well adapted vessel.

The triune essence of mankind created in the image of God is one of spirit, soul, and body.

The bible quotes God as telling Adam that on the day you sin, you will die.

?But of the tree of the knowledge of good and evil, thou shalt not eat of it: for in the day that thou eatest thereof thou shalt surely die. Gen 2:17

Adam sinned in the garden, was sent out, yet lived another 900 years or so. What died? Not his soul. Not his body. It was his spirit.

This is why Christ taught you must be born again. Nicodemus asked how can I be physically born again, but Christ informed it was your spirit.

Jesus answered and said unto him, Verily, verily, I say unto thee, Except a man be born again, he cannot see the kingdom of God. John 3:3

That which is born of the flesh is flesh; and that which is born of the Spirit is spirit. John 3:6

So, to answer the OP, the triune nature of man reflects the triune nature of God.

The Image of God is a very important biblical topic, but what exactly the scripture's authors meant by it is debated. Being made in the Image of God has come to be seen as the defining characteristic by which humanity is distinct from the rest of creation. More than any animals, even more than the angels (who may well surpass humans in intelligence), it is only humans who are like God. We are made like God, and in the incarnation God became one of us, and so humanity is unique amongst all of God's creations.

But what does it mean to be made like God, in his Image? Christians have identified many specific traits which could be part of this Image that Genesis 1 refers to:

• Moral capacity: is a common one. Though animals may kill, only humans can murder. As God declared at the end of day 6, all of creation was "very good", but only God, angels, and humans can be righteous or wicked.
• Intellectual capacity: humans have the capacity to perceive and understand the world, and to recognise the signature of God in what he has created.
• Relational capacity: like God, humans are inherently relational beings, just as it was not good for Adam to be alone, so too are relationships important for all humans. Without them we suffer or even die.
• Communicative capacity: humans, unlike all other earthly life, have the unique capacity of language.
• Creative and artistic capacities: like God we have the ability to be creative, and we can recognise and enjoy beauty.

Some of these would be recognised by all branches of Christianity, but some are particular to multi-person views of God (Trinitarianism and Binitarianism). In the Trinitarian understanding of God, God is inherently and eternally relational and communicative. These are not just traits that could be exhibited by God if he decided to create, but the constant and permanent nature of God. And if God had not created the universe (as almost all Christian branches would say he was free too), then would it really be right to say that God is either of those things if they were but unrealised potentials? For the Unitarian while God may now be relational, he cannot be inherently, eternally, and uncontingently relational like he is for the Trinitarian.

God is likewise the ultimate creator, but we would not say that he was eternally creative, for before he had begun creating, he had not been creating. But we can say that his creativeness is analogically related to what we believe to be one of God's eternal attributes: his begettingness. The Father begets the Son eternally and without beginning, and so it is natural that this eternal attribute of God be expressed in his temporal creation of the universe.

Many of these traits would seemingly apply to angels as well as humans, at the very least moral, intellectual, and communicative. But we teach that angels, mighty beings though they are, were not created in the Image of God. Some of these traits might help us understand the ways in which humans bear the image of God more than angels, such as our relational and creative capacities. I'm not sure the scriptures ever describe the angels as being truly relational, and though they are God's servants, I don't think they're ever called the friends of God (though Job does call them the sons of God...) Likewise I'm not sure the scriptures ever describe angels as being creative. Though they may be musicians in the heavenly court, is that a creative expression of their own, or do the follow the creative instructions of others? Hard to say.

But there is one last understand of the Image of God which I think is most helpful for understanding our image-bearing to the particular triune nature of God. This is actually the dominant understanding for many Protestants (and possibly others): bearing the Image of God refers to our vocation of representing God in this world and being his delegated rulers of the world. This view has the benefit of most closely being exegetically tied to the passage, as verse 26 directly refers to our role of ruling the creation. It is also supported from extra-biblical history, in which many other nations taught that their kings were the "images" of their Gods, possessing divine mandate to rule. But in the Bible it is not merely the king who has this divine mandate, it is all of humanity, both male and female. Here too we separate humanity from the angels, for they were created as God's servants, but not to rule over the earth. We are to be, collectively and individually, representatives of God, living exhibits of his truth, goodness, and beauty.

And here we tie it back to our understanding of the Trinity. For although we teach that all three persons of the Trinity were involved in creation, are involved in sustaining the universe, were part of God's self-revelation to the world, participated in salvation, and the Church is united to all three persons in faith, there is one divine person who is the particular focus of God's self-representation. It is the Son in particular who is God's representative to humanity. It is the Son who is the ultimate communication from God to humanity, and who in particular will directly rule humanity as our King. While we are each made in the Image of God, it is Jesus who is in a different sense The Image of God (Hebrews 1:3, Colossians 1:15, 2 Corinthians 4:4, etc). Although the Son took on human flesh on in the incarnation, there is a very real sense in which we are made in the image of Jesus Christ the Son. As humanity has failed in its vocation of being God's representative rulers of the earth, as the Church is the redeemed people of God, adopted as God's children, we can now represent Christ to the world. Christ is the new humanity and has taken up the vocation to be the Image of God that we abandoned in sin, but as he redeems us we can now participate in him, and once again be the Image of God not just in our human capacities, but in our divine-given vocation.

The image of God is the One come in humanity - Christ Jesus.

Of Adam, Paul says that he is :

... the figure of him that was to come. [Romans 5:14 KJV]

Adam is just a figure of the One who, 'from before the foundation of the world' was the One in whom God's people were chosen ; the One who would come, who must come, who was promised to come.

It is not Adam who is the image of God. And the first account of Creation (the 'Elohim' account) is not to be taken as anything other than a spiritual account of creation as such that is to say as the entire creation (including the new creation) seen objectively.

The second account of creation (the 'Jehovah' account) is that which places Adam and Eve in a garden of providential provision in order for events to unfold.

Natural humanity (Adam and Eve) set forth a created picture of Christ and the Church :

For this cause shall a man leave his father and mother, and shall be joined unto his wife, and they two shall be one flesh. This is a great mystery: but I speak concerning Christ and the church. [Ephesians 5:32 KJV]

Adam and Eve do not set forth the 'image of a triune essence'. That idea is altogether false and is nowhere stated in the documents of the Council of Niceae (or anywhere else I can think of).

It is the Son of God, come in humanity ('come of woman' and 'come under authority'Galatians 4:4) who is ...

... the brightness of his glory, and the express image of his person, and upholding all things by the word of his power, when he had by himself purged our sins, sat down on the right hand of the Majesty on high; [Hebrews 1:3 KJV]