It looks like the NYT article is taking from this CBO report: https://www.cbo.gov/system/files?file=2018-06/53919-2018ltbo.... You'll notice that in the article and in the report, it's shown as "Net interest" which is defined by the CBO as the amount the US government pays out in interest payments minus what it takes in interest payments (CBO glossary: https://www.cbo.gov/sites/default/files/114th-congress-2015-...). So the interest payments on the ~40% of the debt the goverment owes to itself would cancel out.
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.
... 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. ...
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. ...