I believe a .410 slug is just what it says, .41. It has rifling and a skirt built in to seal the bore, and it's soft enough that it will swage down to full choke guns of about .391.
Some guns that are chambered in .45 long colt will also fire a .410 shell, but have a choke adapter, like the T/C contenders. I have also seen derrengers that are suposedly .45LC/.410, but they don't have a restriction for the shotshells, they just bounce down the barrel.