Not at all a misrepresentation of the classical Trinitarian doctrine!
I know how the doctrine goes, but the doctrine is not based in reality, making a contradiction, that no one can explain, because three persons make three Gods, not one God, for..
Three persons judging, whether they all be in one body, or not, whether they all agree, or not, are still three judges judging!
The Father person is God all on His own, the Son person is God all on His own, the Holy Spirit person is God all on His own. Three persons that are God all on their own are three Gods, ..but magically the Trinity doctrine says, "No, they are not three Gods, according to our doctrine!"
And if you mean the word "God" means essence, as in: "
God's personal characteristics, or to the facets of His personality. Sometimes the term “Attributes of God” is used to refer to God's essence," then three separate characteristics, three separate facets, and three separate attributes are still three Gods.