Empathy is the ability to understand someone else’s experience, to grasp something of what they’re thinking, feeling, etc.
It doesn’t automatically imply that you care, you can respond to that understanding of someone else’s life with compassion, indifference or anything else.
The colloquial usage of the word empathy to mean “consideration and caring” is problematic as is oftentimes an imagining of how the observer would feel if they were in the difficult situation, rather than the useful version of understanding how the other person feels within that difficult situation.
Speaking from a therapist point of view…
Empathy is the ability to understand someone else’s experience, to grasp something of what they’re thinking, feeling, etc.
It doesn’t automatically imply that you care, you can respond to that understanding of someone else’s life with compassion, indifference or anything else.
The colloquial usage of the word empathy to mean “consideration and caring” is problematic as is oftentimes an imagining of how the observer would feel if they were in the difficult situation, rather than the useful version of understanding how the other person feels within that difficult situation.
My understanding is that they refer to different types of empathy.
What’s you’re describing is cognitive empathy and in the OP it’s describing emotional empathy.
https://www.verywellmind.com/cognitive-and-emotional-empathy-4582389
https://dictionary.cambridge.org/dictionary/english/empathy