That's because mafia refers to itself as "Cosa Nostra", "'Ndrangheta", "Camorra", and "Sacra Corona Unita" depending on where in Italy is originated from.
"So Paramount agreed and [the League] thought it’d won this great victory despite the fact there was only one use of ‘mafia’ in the script.”
Doesn't "Cosa Mostra" translate to "this thing of ours"?
To add to OP, mafia or cosa nostra is just Sicilian, ndrangheta is Calabrian, camorra is from Naples/Campania.
Usually the word mafia is used worldwide to refer to organized crime in general, but mafia there is just one.
If you ask a person from Naples if there is mafia there, they'd say: we have no mafia, we have camorra.