When I see total US public debt, I wonder whether state debt is included. If "total US public debt" used differently for different cases, I will restrict my case to the usage in national accounting/St. Louis Fed FRED data.
No. The figures commonly cited as the US national debt specifically exclude state and local debt. State and local government do hold part of the US national debt either purchased or issued to them through the State and Local Government Series Securities program, but the St. Louis Fed figures don't include debt issued by states and local governments, e.g. municipal bonds.
The US Treasury Department defines the public debt as
as public debt securities issued by the U.S. Treasury.
The US Treasury doesn't issue debt on behalf of states and local governments, so the public debt figures exclude this. If you want a detailed breakdown of what comprises the public debt, look no further than the Daily Treasury Statements, specifically Tables III-A, III-B, and III-C.
Side note: The "Fed" isn't an acronym. It's just an abbreviation for the Federal Reserve System. The St. Louis Fed's FRED system, however, is an acronym (for Federal Reserve Economic Data).