Absolutely, I'd edit the post but rather that acknowledge the gaff here. Debt the government owes to itself it has a number of ways to manage, debt between private entities is a much harder thing to fix without tanking the economy. The canonical example these days is the subprime mortgage crisis.
"Public debt" is another name for "Government debt"[0]. From the rest of your description, you means "debt owed by the non-government public", private debt, or "household debt" (which doesn't strictly include corporate debt like your example outlines).
[0] https://en.wikipedia.org/wiki/Government_debt