That is, Peter, the classic distinction without a difference.
"One nation, under God" still requires a thought as to which God and whose God.
Otherwise, we would say "one nation, under (we think) some God somewhere (if there is a God), indivisible, with liberty and..."
It is not a distinction without a difference. Unlike Imperial Japan, Nazi Germany, or the Soviet Union, we insist our state is at best penultimate.
For about 1/2 of the 129 years since the Pledge of Allegiance was created, "under God," was not part of it. It was added in 1954 to show that America was not like atheistic communist Russia.
The three different versions are listed below.
Original: 1892 Francis Bellamy
"I pledge allegiance to my Flag and the Republic for which it stands, one nation, indivisible, with liberty and justice for all."
1923
"I pledge allegiance to the Flag of the United States of America and to the Republic for which it stands, one nation, indivisible, with liberty and justice for all."
1954
"I pledge allegiance to the flag of the United States of America, and to the republic for which it stands, one nation under God, indivisible, with liberty and justice for all."