You have at least two definitions / conceptions of God then, (1) the Father, (the non-Triune God), and (2) the Triune God.
But what about God the Son and God the Holy Spirit? They are also non-Triune Gods "in themselves" per your philosophy.
As Servetus might have reasoned, in analytical terms, Trinitarians actually have around 4 or 5 definitions / conceptions of God, as the substance of God can also be construed as a fifth God in its own right, as imparting deity to each of the other members of the Trinity.
BTW, the bible only has one definition / conception of God: the Father, who confers divinity/deity on the Son and the Spirit.