Hopefully some useful Trinitarian reflection about Trinity beliefs:
I think that what matters is that the interpersonal relationship between the Father and the Son is genuine and not a performance by a single self...
I agree. The error in question is Modalism, which arose in the late 2nd century – by such persons as Noetus of Smyrna, Praxeas and Sabellius, and is still around today, as per Oneness Pentecostalism.
The Trinitarian (theologian) would also assert, as prior, the
intrapersonal relationship between the Three Persons of the Trinity, traditionally expressed as the difference between the 'Economic Trinity' (from the Greek οἰκονομία,
oikonomia, 'household management'), and the 'Immanent Trinity'.
The Economic Trinity refers to God’s external actions and roles within creation and salvation history, specifically how Father, Son and Holy Spirit interact in, with and to the world. The Immanent Trinity (or Ontological Trinity) describes God's internal, eternal relationships and nature as Father, Son and Holy Spirit, existing independently of creation and human experience.
The Economic Trinity focuses on God for us in time and place; the Ontological Trinity focuses on God in Himself in eternity. The former serves as the revelation through which humans come to know God, whereas the latter represents the transcendent reality that remains incomprehensible in its fullness.
... and that Jesus has all the authority of God, from God.
Whereas a Trinitarian would suggest Jesus has all the authority of God because He is God.
I disagree with any theory that makes same-essence belief a requirement for his purposes.
In which case you reject a core axiom of the traditional expression of the Trinity.
I think that he is in some way a perfect reflection of God ...
As above, I would suggest Jesus is a perfect image of God because He is God. (I think, technically, He is the image and we see its reflection? I might be splitting hairs.)
... and that is part of his nature unique to him, not possible for anyone else.
You'd have to qualify that, as neither the Son's divinity nor His humanity is, in itself, unique. His divine nature is the same as the Father's and the Holy Spirit's, but neither the Father nor the Holy Spirit is the Son. His humanity is common to us all, in that his human nature is the same nature as ours. He is unique as the Son, the Second Person of the Trinity, as He alone occupies that place in the intrapersonal relationship, and He alone occupies that place in relation to the world.
He's fully human, but something else at the same time, which I don't think is possible for anyone to understand ...
You can declare that for yourself, but you cannot declare that for everyone else.
... or that we need to understand other than his authority being God's authority ...
That, as explained above, is a misconception when applied to the Trinity.
the very meaning of the authority of God in the Abrahamic scriptures.
Authority of God rests in God and reaches down all the way through creation and is invested in whosoever God chooses for whatsoever is His purpose – patriarchs, prophets, priests all had a divine authority invested in them to speak and to act, and so on.
If same essence means same self, or same substance, then it's false.
Well you've offered two possibilities here.
'If same essence means same self' – it does not, you are right, it is false.
'If same essence means same substance' – it does, so you are wrong in declaring it false.
