How do you differentiate between a "classical" gold standard and a gold-exchange standard? I believe Austrians prefer the former in which money is not "fixed" to a particular price of gold but defined as a specific weight of gold, and the price floats.At the level of constitutional monetary reform, introducing either a "classical" gold standard or a gold exchange standard involves choosing a price of gold. To say that one dollar is a certain weight of gold, say, a dollar is 1/200 of an ounce of gold is the same thing as saying that the official, defined price of gold is $200 per ounce.
Changes in the weight of gold that defines the dollar are equivalent to changes in the dollar price of gold. Such changes (devaluations or revaluations) could be constitutionally prohibited, or could require special procedures--supermajorities of some sort.
Changing the unit of account to gold ounces is also possible, but at the level of constitutional reform, the shift still involves choosing a dollar price of gold in order to shift from the existing dollar-denominated world to the new system. If the "gold ounce" unit of account were introduced, changes in the price or weight of gold would similarly require the introduction of a new unit of account, though a special "ounce" applying to gold only would be possible.
People should be free to introduce alternative units of account, like gold ounces, and quote prices, keep accounts, and make contracts in terms of those alternatives if they want. Because of the advantages of using the same unit of account everyone else does, this freedom is unlikely to have much impact. Only if a Zimbabwe-like hyperinflationary disaster occurs, could a gold standard evolve in the midst of the wreckage. This would be an example of getting rid of all existing money and waiting to see what evolves.
The live issue is constitutional monetary reform--a shift to a new definition of he dollar. Commodity standards, like a gold standard, are possible candidates. But introducing a commodity standard involves choosing an initial dollar price of the commodity.
Most Austrians oppose having a monetary authority or central bank issue monetary instruments and instead favor leaving that to private banks. Still, if a monetary authority issues dollar-denominated currency or deposits, making those monetary instruments redeemable in the amount of gold that defines the dollar is a way of enforcing the requirement that the market price of those instruments remain at par. That is equivalent to having the paper money price of gold remain equal to its defined price.
That the gold price of paper currency denominated as one dollar cannot be 50 cents is the same thing as saying that the paper currency price of gold cannot be 2 paper dollars per gold dollar.
With free banking, it is the same. If each bank is obligated to redeem its dollar-denominated currency and deposits with gold, that is a way of enforcing the obligation that the market price of its monetary liabilities remain at par. This is equivalent to the price of gold in terms of those notes or deposits remaining at its official, defined price.
My perspective is influenced by awareness of the various interventions that gold bugs/bullionists/ merchantilists used in the early modern period to encourage gold coin and gold reserves. Reserve requirements are the most blatant, but the ban on the option clause and restrictions on small denomination banknotes had the same goal. The "classical" gold standard was rife with regulations that prevented movement towards what would amount to a privatized gold exchange standard.
Regardless, the relative price of gold depends on the supply and demand for gold, including the demand, if any, for monetary purposes. With a gold standard, the price of gold is fixed by definition, and so the price level must adjust to clear the market for gold. The banking system, issuing gold-denominated monetary liabilities, must restrict both the total quantity issued and the amount of each type to the demand to hold them in order to keep the prices of those monetary liabilities at par. Changes in the amount of gold demanded for monetary purposes has the same macroeconomic effect as changes in the demand for gold for use as jewelry. If governments hold stocks of gold, then changes in the government demand for gold can have macroeconomic impacts.
For example, suppose in the aftermath of World War I, a large proportion of the world gold stock was held by the Federal Reserve. How the Federal Reserve chose to manage that stock would have implications for the equilibrium relative price of gold and so world macroeconomic conditions. It is unclear to me how much difference it makes whether such manipulations occurred under a classical gold standard or a gold exchange standard.