There are so many experiences that all get collected together and off-handedly referred to a "empathy" conversationally. The first is what researchers call affective empathy -- feeling another person's feelings, so to speak, and the experience that most likely first comes to mind when a person hears the term.
Empathy isn't ever really feeling another person's feelings, though, as no human is psychic and the ways that we signal emotions vary from person to person and across culture, gender norm, age cohort, and much more. So at best, we can call that kind of empathy an intense internal simulation of what we believe another person to be feeling. From all that I've gathered, to the people who experience affective empathy, it's message is seductive -- it feels like the truth of another person's internal state, and it feels very important. But as someone who is frequently mis-perceived by high empathy individuals, it feels like a distracting, misleading, intuitive way of processing that is not rational or consistent and not to be relied on for any important judgement. But it might make enjoying a movie more fun.
The second kind of empathy is what psychologists call cognitive empathy or perspective taking. This is the reasoning out of a person's emotions based on things like our own prior experience, things that we've read, and our own logical gaming-out of how we might feel or how it might make sense to feel. This is something that can be practiced and honed as we learn more about other people. It can also include some pattern-matching behaviors, such as manually recognizing that certain facial tics or postures often signal certain emotions. This sounds pretty similar to several experiences you describe in your piece. Because this kind of empathy is intentional and can be honed, it seems to me a lot more useful for guiding respectful, compassionate behavior.
Sometimes, people use the word empathy simply to refer to the act of being compassionate. This really muddies the waters, I think. Our behavior is a choice -- our feelings and even many of our thought processes are not. Even if a person can't feel other's pain and finds it hard to game out why somebody might feel as they do, they can still elect to be patient, inquisitive, a good listener, giving with their resources, and interested more broadly in making the world a better place. (I guess we technically could also divide compassion into the micro and macro levels -- caring about our close associates sure seems very different from caring about the world, and some people are better at one of the other it seems to me).
Finally, another internal experience that might be mistaken for empathy, or might be wrapped up in empathy, is one's arousal level in response to gore, blood, self-harm, descriptions of crime, etc. I think in many cases stress arousal is linked to affective empathy, as you describe them being linked for you. But I think they can diverge. So for instance, hearing a sad story about a person losing their family farm might not move me emotionally, since it's not something I have experienced or place too much weight in. Cognitively I can recognize why someone would be very upset to go through that experience, but hearing about it I can't really "feel it". I also can't "feel" really extreme drug addictions, or other compulsions I haven't had. But I get very physiologically distressed by detailed descriptions of blood being lost, self harm, puking, and other intense physical experiences. And I even kind of imagine those experiences happening to me when I hear someone else relate them. All while not usually having affective empathy. It's more like a visual plus a head rush and a stomach churn more than like imagining the pain. It seems likely to me that many people who have endured trauma become either way more reactive to physical pain/gore stimuli, or they become numb to it -- and horror is popular I think often among the people who become numbly fascinated by gruesome things.
Anyway, all this is to say is that it's a very gnarled tangle of experiences, relating to other people and caring about them, and that we do humanity a disservice when we pretend that the only route to behaving well toward others is via simulating internal feelings of their pain or sadness. I think both you and I are object lessons in that not being the case.
