That sounds like Modalism to me, which was an early heresy. I don’t understand why the need for analogies arises among christians – and I don’t mean that to be rude. The Trinity is unlike things you can see or hear in the natural world [like deer], so what’s the point of trying to ‘water it down’, as it were?
I don’t understand the practical purpose of it. I’ve heard the water, ice and steam analogy, the egg analogy, the apple analogy, the cerberus analogy [the worst one, in my opinion], but none adequately describe the three Persons and their relationship with one another; and at the same time how these persons are one in Essence. It doesn’t work for me.
Your contradiction and confusion begins with the difference between God’s Essence which never comes down to us, which both Catholics and Islam teaches and the presence of God which comes down to us the divine revelation of God in Trinity of persons.
Faith is a gift from God, faith supersedes the carnal mind and understanding of the natural order of things, for the faith God gives reveals from which all things came into being from the spiritual eternal reality existing, while the old order of things are close to disappearing.
God has revealed to us through the Spirit. For the Spirit searches everything even the depths of God…no one comprehends the thoughts of God except the Spirit of God. Now we have received not the spirit of the world,** but the Spirit which is from God, that we might understand the gifts bestowed on us by God. And we impart this in words not taught by human wisdom but taught by the Spirit, interpreting spiritual truths to those who possess the Spirit.**
The unspiritual (natural) man does not receive the gifts of the Spirit of God, for they are folly to him, and he is not able to understand them because they are spiritually discerned. (1Corinthians 2:6-16)
The subjects (deer, water, ice, vapor) that only reflect what the intellect can grasp at the Trinity they never define the Trinity. These describe spiritual realities described in spiritual terms that God reveals from His Spirit to our humanity, that parts to us the gift from God, Faith.
Because the blessed Trinity reveals the True presence of God such as in the burning bush, whispering wind, rock, fire in the desert at night and cloud in the day etc… the True Essence of God does not come down to our humanity. Although the three persons of the blessed Trinity are con substantial in True divine Essence one God, the Father, Son and Holy Spirit are distinct in person and revelation of God to our frail humanity.
We do not call the rock, fire, cloud in themselves God, we believe God revealed His presence from these, when God our Father in the fullness of time, reveals His presence in the person of the Son and in the person of the Holy Spirit all three distinct from one another, True faith does not limit God to what God can do, for God can do all things, and nothing is impossible for God, when God reveals to our Spirit all three distinct persons in the blessed Trinity is One God.
Jesus has revealed to our humanity, “The Father and the Son are One”, “If you have seen me you have seen the Father”. Our human analogies of the One God in Trinity falls short every time of Jesus own teaching and revelations of God’s presence in the Trinity.
Peace be with you