"The main rationale behind the order was actually to remove the constraint on the Federal Reserve preventing it from increasing the money supply during the depression. The Federal Reserve Act (1913) required 40% gold backing of Federal Reserve Notes that were issued. By the late 1920s, the Federal Reserve had almost hit the limit of allowable credit, in the form of Federal Reserve demand notes, which could be backed by the gold in its possession"
One is a bank defaulting on its obligations to provide you a certain number of units of currency regardless of their market value. This has not happened since the FDIC was created in the wake of the Great Depression.
Another is a sovereign default where the country decides not to honour its obligations to those holding units of its currency or its debts. This has happened ~6 times in the history of the United States, and I do include the cancellation of the gold withdrawal rights as a sovereign default. This has nothing to do with the FDIC or your bank.
I can't stress this enough, neither of these things matter to you if you do what you're supposed to with money, which is not keep your life savings in a mattress or a savings account, but use it as a temporary store of value, medium of exchange and unit of account and buy things with it. Gold, real estate, magic beans, I don't care. This is practically ECON-101.
On the subject of freedom, backing all currency with gold eliminates my right to choose what I back my personal economy with. A gold backed currency has strictly less freedom than a fiat currency. I can today choose to back my personal economy with gold, silver, platinum, magic beans (crypto) or Apple stock by leveraging a brokerage account and a credit card. If you believe in freedom, it's actually what you want.
Executive Order 6102 is an executive order signed on April 5, 1933, by US President Franklin D. Roosevelt "forbidding the hoarding of gold coin, gold bullion, and gold certificates within the continental United States." The executive order was made under the authority of the Trading with the Enemy Act of 1917, as amended by the Emergency Banking Act in March 1933.