"One nation under one God"<quoted text>
All that's saying is that a large number of people in the U.S. are Christian.
It isn't a declaration the U.S. is a Christian nation in any legal capacity or official sense.
"In God We Trust"
and others are all declarations.
Also as far as I know a Supreme Court judge, and also the US President, is required to take an oath on the Bible before being appointed.
It is the same in US law courts, where one must take an oath on the Bible before standing for witness.
So the Bible and God are foundational in the US legal and Constitutional system.