What actually is psychopathy?

So thought I would pose this question to fellow INTJs, since we are often cast as the villains in fiction. First off, I do not have a background in psychology, so I would appreciate any corrections or clarifications from someone more informed than myself.

That being said, I'm honestly confused by the clinical definition of psychopathy. Half the time, it just sounds like someone who lacks impulse control, but then sometimes the definition also includes a deficiency in affective empathy. Are both needed or just one or the other?

Also, I don't get how deficiencies in either of these lead to violent behavior. Just because someone is apathetic towards the thoughts and feelings of others, and doesn't stop to think things through does not mean they want to harm others. There is missing motivation there. Where does that motivation come from?

And circling back to the affective empathy thing in particular, I don't see how this deficit leads to antisocial behavior, especially when considering that psychopaths are very proficient in cognitive empathy. If you lack affective empathy, but are proficient in cognitive empathy, couldn't you just as easily be prosocial, since realizing that a flourishing society also benefits you? Hence, you would follow basic rules and reciprocate various kindnesses? Again, it just seems like there is a missing piece here.

Admittedly, this topic frustrates me, and perhaps I'm projecting, but I'm guessing others (read "most") with our personality type because I do (and maybe we all) have deficient affective empathy and proficient cognitive empathy, yet I (we) don't exhibit antisocial behavior. I for one certainly don't want to hurt anyone (including nonhuman organisms other than for survival), and I would also prefer a world that is kinder. Hence, I try to practice kindness. (Though, I do get pretty salty around falsehoods, irrationality, and injustice.)