My example was off the mark now that I look at it, but your statement is equally flawed.
A sociopathic killer would be one that kills through careless and irresponsible behaviour, it's just there's no use for such definition as it's already suitably covered by manslaughter. A better political example might be: Boris Johnson is a sociopath, and as the result of his careless and irresponsible behaviour many more people died during the pandemic. Meanwhile a psychopath might be seen as someone who intentionally kills with a similar disregard.
So, to fix my example, Boris and Trump killing through reckless policy would be sociopathic killing, while Trump killing his political opponents is probably more psychopathic killing.
Technically though, there's no such thing as a psychopathic killer either, as in clinical terminology both psychopathy and sociopathy refer to people living with Antisocial Personality Disorder.^1 Most mental health professionals won't differentiate between the two nor use either as a diagnosis, but some psychologists or researchers consider psychopathy as an extreme form of Antisocial Personality Disorder.^2 However the actual diagnosis used these days is apparently Antisocial Personality Disorder.