A depositor is considered an "unsecured creditor" to the bank. What I get from the doc is that as an unsecured creditor, if the bank goes under then the depositor funds will be taken to pay secured creditors. The scam though is that the depositor has "deposit insurance" provided by the government for up to the limit amount. The problem of course is that the deposit insurance is itself "unsecured" and based entirely on the "full faith and credit" of the government. As anyone old enough to remember the S&L crisis of the '80's can attest, the depositors who were insured by the FSLIC were lucky if they got $0.40 on the dollar of their so-called "insured" deposits. Of course, the uninsured deposits were a 100% loss.
We don't hear much about how the FSLIC failed to actually protect people's deposits. If there is a general bank failure in the US, my guess is that the payout by FDIC will be much less than $0.40 on the dollar. FDIC is insurance in name only.