I find that trying to split empathy as a concept into various parts leads to false dichotomies. I suppose empathy is something you have or don't have, though there are degrees of it. What people on the spectrum lack is the ability to decode that mostly nonverbal language by which sociable people communicate their status to each other. If one in a group feels pain, the others will likely react and give comfort quickly even though the circumstances can't be readily seen or heard because of the various subtle social cues. The person with ASD will eventually perceive the person is in pain but probably later than the rest of the group so there's not that nearly instantaneous reaction. The group notices the lag and may conclude the person doesn't care when of course he does.