I'm not sure where Damien is getting his info, but the cutlass is a 17th C. invention and was widely used during the Golden Age of Sail. We have contemporary illustrations of pirates pictured with cutlasses. Pirates and other corsairs would've also used hangers, messers, and any other relatively short swords with broad, slashing/chopping edges.
The Golden Age of Piracy was roughly from 1650-1730. They definitely used a weapon that many of them called 'cutlass' - or some word essentially translating to something akin to 'cutlass.' But that, in itself, is not particularly helpful here. We have a similar problem with all kinds of sword names and how they were used or evolved over time versus the common pop culture understanding. For example, I would say MOST people think of a 'longsword' as a single-hand knightly sword. Even though that's not how we use the word... academically, if we can use that term.
What people tend to think of as a cutlass, and therefore what gets associated with early 18th century pirates in media, is absolutely a cutlass. But it's a 19th century design. What an 18th century person would have called a cutlass almost certainly is not what people here in this thread would think of if you said 'cutlass.' 18th century cutlasses were essentially hangers, maybe what some people here would recognize as a form of 'saber.' Or dussack, if you're nasty. And they would have had very different blade types and hilt furniture from the later 19th century cutlass we tend to associate with pirates.
So when I say that pirates didn't use cutlasses, I mean the cutlasses most people see in their minds when they hear/see that word, rather than saying pirates didn't use the word 'cutlass.'
Also, it's not true that pirates would have used short swords with broad, cutting blades. Not exclusively at least.
What would be a historically accurate sword for a pirate?
For a pirate of the 'Golden Age of Piracy' -- the easiest and also most complicated answer is 'whatever was around.' That is to say, whatever swords were commonly available to that particular person. As I said above, what you probably see in your head at the word 'cutlass' is from the 19th century. So.. somewhere around 100 years after Blackbeard eats it. But if you look up the types and styles of sword available around the world between the mid-1600s to the mid-1700s... that is the answer. All of them. Things we might call rapiers, sideswords, dussacks, hangers, broadswords and backswords. And, of course, katana, dao, jian, nimcha, and various types of swords most people here might recognize as near-Eastern "scimitar"-type swords.
It likely depends a lot more on the individual pirate's ethnic background, wealth and status, and personal preferences. What they didn't have was time travel. So a 19th century cutlass is about as likely to be in an 18th century pirate's hands as a Colt 45.