Nope. We have interpreted it correctly.
The Father is not the Son.
"Essence" speaks of the Godhead. In the unity of the Godhead there are 3 persons (hypostases, if you will.) These three persons are Father, Son, and Holy Spirit.
The one divine essence refers to the Godhead.
The path is full of thorns.
""Essence" speaks of the Godhead. In the unity of the Godhead there are 3 persons (hypostases, if you will.) These three persons are Father, Son, and Holy Spirit."
Part of the problem is in using the word "person" for hypostasis; that's not what it means and its always a mistake to attribute human characteristics to God.
Try looking at "Against the Arians" by +Athanasius the Great. It is a marvelous work on the Son in relation to the Father and the Trinity. It is thoroughly "orthodox" and in sync with the Creed. Its very long, but there is a pretty good paper on it presented at a Baptist seminary which you both might find interesting (pdf format):
http://www.templebaptistseminary.edu/alumni/jones.pdf.