The way it works is a little bit more complicated than simply choosing "wanted" ships and destroying them, hence the offers for many different target types.
On the left hand panel, the contacts tab will give you more information about your target. Do they belong to the faction that runs the system? If so, attacking them will turn that entire faction temporarily hostile (i.e. until you jump or log out), and normally the security forces in the system will belong to the controlling faction.
Someone who is wanted isn't a pirate by necessity. There is typically a faction whose members are smugglers and pirates (e.g. a mafia), but smugglers won't count as pirates for kills. Other factions will also have pirates, it's just quite a bit less likely that they are ACTUALLY a pirate, and not just wanted for something else.
Try to identify pirates by their behaviour. Do they cargo scan you and taunt you? Do they have skull decals? The job is specifically asking you to kill NPCs who perform acts of piracy.