Cognitive empathy is something I am learning because I got out of a destructive/manipulative relationship. I have healed a bit but have more work to do. As I have gotten better I have asked myself questions on how I couldn't se the manipulation all along. Like the red flags were there from the very beginning and I didn't se them.
So, since about a year back I am learning cognitive empathy because I have come to the conclusion that
1.I might have a difficulty reading people as a result of manipulation, gaslighting and abuse (I have PTSD) or
2.I didn't understand cognitive empathy from the beginning witch made med vulnerable to begin with (I have always been gullible as I believe the best in people). And I have never understood when guys have liked me in the past or how to approach them. I believed in what you see in Hollywood movies (yes, that was my view of love).
So, to answer your question, because I am still trying to figure out if I showed signs of being on the spectrum as a child (so far my father has mentioned that I learned the alphabet when I was two, which he found unusual, I wanted to be home alone from school and learn to take care of myself when I was 7, like Mathilda from the book by Roald Dahl, and my mum was often annoyed that I was so easy to...well manipulate. Even as a child with other children I apparently always did what other people wanted, even if it resulted in me getting into trouble. My mum mentioned that I often "rather" hung out with other families than my own, I don't remember that and I don't remember disliking my own family so...I don't know). Anyway, if it is a result of the abusive relationship, PTSD or a sign of being on the spectrum, I still need to learn cognitive empathy.
I started to learn from this guy on Youtube a year ago, the channel is called Asperger from the Inside. That guy has some really good ways of communicating cognitive empathy and explain things in ways I understand.
But the best way for me to communicate is honesty and verbal communication.
As far as cognitive empathy goes I know that crying can be out of happiness, because one is sad or angry (here facial expressions is key).
I know that something that looks like anger can be the result of fear or anger (here I always have to ask, I think an angry person prefers to be left alone and a scared person might need comfort).
I know that being sad can make people feel vulnerable and that might want to be something you want to hide (the vulnerability I mean). So a sad person might say: "I'm ok!" but the tone of voice can be "flat" and the gestures are not the gestures of a happy person (like a sad person, I think someone who is sad can look down to the ground more etc.).
A happy person...my god. More difficult. Because I know people use happiness to hide difficult things.
But cognitive emotion is difficult and it is different from one person to another I guess.
Since I work as a teacher with children, I teach these things and it is difficult. I don't understand my own reactions at times.
What helped me with cognitive empathy is that Youtube channel I mentioned but I also have a small group of people who I trust and I ask. I can ask: "The words you are saying are positive (I am interested in languages and words) but you look somewhat angry", are you happy or angry or something else? And those close people have helped me understand a lot. I have a couple of people I can ask at work, a close group of friends, my boyfriend and my parents and sister. My sister and I can read each other.
So this is what I've got, I hope you can find something useful in my answer.