Chemistry - Dependency of Standard Gibbs free energy (using electrode potential) on stoichiometric moles of electron exchanged

It does not depend on how you write the reaction. You were right in saying that, $n$ is the mole number transferred in the reaction. So, it is not the number of electrons you write arbitrarily in the equation that represents the reaction, it is rather proportional to the number of other substances, which, even being all arbitrary always satisfy that proportional relation.

Think of $n$ as the factor needed to relate one intensive property with one extensive property.