So this is more or less right but not precisely accurate.

Psychopathy and sociopathy do name the same condition, earlier names for which include Pinel's "insanity without delirium" and Prichard's "moral insanity". And the clinical community did favor "sociopathy" to "psychopathy" for a time, and the reason for the change was to emphasize the dimension of "nurture" over that of "nature", so to speak. But I don't think psychopathy and sociopathy were clinically distinguished based on mode of symptom acquisition, I think the idea was rather to replace "psychopathy" with "sociopathy" to emphasize the learned component (for "political" reasons, as it were). Like most conditions, we now know that both dimensions play a role (there is a heavy genetic load, but there are usually also identifiable environmental factors).

Anyway, that's minor and partly speculative stuff (about the intention behind the temporary dominance of "sociopathy" over "psychopathy"); the real reason I'm writing this reply is what you said about ASPD.

In the clinical community of psychiatry, if we take this to strictly follow the DSM, there is only ASPD. But in the broader community of psychology (and psychoanalysis, for example), "psychopathy" is still a completely valid and widely used clinical term ("sociopathy" has fallen out of favor as "psychopathy" did for a time). Also in certain non-medical communities, such as law enforcement and prison administration (where Hare's work, and especially his diagnostic checklist for psychopathy, the PCL, find much use as well as empirical validation).

But the important point is that ASPD and psychopathy are distinct clinical constructs. The DSM just doesn't categorize psychopathy any more, they find ASPD sufficient for their purposes—but that doesn't make them the same thing. In particular, there is this crucial difference: someone with ASPD is essentially a psychopath who has been caught. This makes some sense: the DSM is a tool for evidence-based diagnosis, meaning there has to be evidence (e.g. instances of juvenile detention, arrests, etc.) in order for there to be a diagnosis. The evidence requirements for ASPD in particular are fairly heavy/strict, because a diagnosis of ASPD is much more socially important than a diagnosis of say depression or even B-cluster personality disorders like Borderline.

tl;dr: psychopathy is a well-defined and widely used clinical construct that is distinct from ASPD; one can be a psychopath and not meet all (or a sufficient number) of the criteria for ASPD.