It's a very bad mistake you are making. The problem is, you should not take whats wroten LITERALLY. What they are trying to describe with the word "empathy" is that they lack "theory of mind" (something about difficulty understanding what others are thinking and feeling). Maybe you could say it is a form of (lacking) empathy. You say "and lack of empathy is the corner stone of sociopathy". But they DONT have the cornerstone of sociopathy more than you. The problem with sociopathy is they have no conscience and guilt about other people. There could be a situation where the sociopaths exploits an other person and the person is hurt, but they dont CARE. They can clearly see the other person is hurt and what goes on in their mind (thats having "theory of mind", what they in the paper call "empathy"), but depite of that they exploit and hurt them and they have no guilt and feeling of caring about them. Thats ALSO called empathy, but autistics have that. There could be a situation where an autistic could'nt see something was wrong, because they did'nt understand what happened in the other persons mind, but that is out of not UNDERSTANDING. Then the person explains it and suddenly they get very sorry for the other person, because they can feel the other person and care. As long as they can see something they care and feel what an other person feels as much as you do.
So the sociopath have good empathy and at the same time bad empathy?
And the autistic have good empathy and at the same time bad empathy?
The explanation here is that "empathy" is just a word for describing something, thats how words function, a word can mean to different things at the same time. But here empathy is describing both one thing and at the same time a completely different thing. Thats the mistake. But it is easy to misunderstand it, if you dont know much about it, because you just think what you normally think about the word "empathy". So it is not because im angry i'm saying this, i'm just explaining it, so you dont misunderstand it.
Another word for what an autistic lack is: Cognitive Empathy or Theory of mind. But a sociopath usually has this.
Another word for what an sociopath lack is: Affective Empathy. But an autistic usually has this.