[nabla9]

Net interest is defined in the terms of federal budget. Fed and many other government institutions are outside the budget.

https://definedterm.com/a/definition/4513 (gives same definition as your first source)

net interest: In the federal budget, net interest comprises the government’s interest payments on debt held by the public (as recorded in budget function 900), offset by interest income that the government receives on loans and cash balances and by earnings of the National Railroad Retirement Investment Trust.

https://definedterm.com/a/definition/4433

... Debt held by the public consists mainly of securities that the Treasury issues to raise cash in order to fund the operations and pay off the maturing liabilities of the federal government that tax revenues are insufficient to cover. Such debt is held by outside investors, including the Federal Reserve System. ...