I'm really confused why there is even an argument... the Fuel Log is quite clear:
It asks for "fuel used". If you want to show that you got your fuel for free, or mixed free fuel with some ratio of purchased fuel, then you show that in the "Price" field.
That's where you show how economical you are from a
price paid standpoint. The "fuel used" field shows how economical you are from
fuel consumption standpoint (regardless where the fuel came from). They are two different measures.
It's honestly not all that complicated.