Circular reasoning.
The value of gold is based on the total demand for gold relative to the supply of gold. Period. The specific purpose doesn't matter.
The form of gold does matter. Coins can have a value distinct from the bullion price of gold since coins and bullion have different utilities.
“The value of gold is based on the total demand for gold relative to the supply of gold. Period.”
That’s exactly right. And the value of a Fed Note is based on the total demand for a Fed Note relative to the supply of Fed Notes. Period.
If you arbitrarily call either “money”, you have forced up the Demand function D for that item. One is a paper fiat, and one is a shiny yellow fiat.