Googling exactly "Costa Nostra" returns essentially nothing specific (150k results vs 4.6m for "Cosa Nostra") and Google tries to redirect it to "Cosa Nostra", so that seems to be a typo.
Wikipedia article on "Sicilian Mafia":
> The Sicilian Mafia or Cosa Nostra (Italian: [ˈkɔːza ˈnɔstra, ˈkɔːsa -]; Sicilian: [ˈkɔːsa ˈnɔʂː(ɽ)a]; lit. 'Our Thing'), also simply referred to as Mafia, is a criminal society and criminal organization originating on the island of Sicily and dates back to the mid-19th century.
Wikipedia article on "American Mafia":
> The American Mafia, commonly referred to in North America as the Italian-American Mafia, the Mafia, or the Mob, is a highly organized Italian-American criminal society and organized crime group. The terms Italian Mafia and Italian Mob apply to these American-based organizations, as well as the separate yet related Sicilian Mafia or other organized crime groups in Italy, or ethnic Italian crime groups in other countries. These organizations are often referred to by its members as Cosa Nostra (Italian pronunciation: [ˈkɔːza ˈnɔstra, ˈkɔːsa -]; lit. 'Our Thing') and by the American government as La Cosa Nostra (LCN). The organization's name is derived from the original Mafia or Cosa Nostra, the Sicilian Mafia, with "American Mafia" originally referring simply to Mafia groups from Sicily operating in the United States.
So you're right that the name with "La" is an US government term for the American Mafia. Funny that the "distinction" was made in the most ambiguous way possible.