There are commonly used terms for different types of empathy[1] which makes these conversations a lot easier. People often generalise, or misinterpret a lower level of one type of empathy for another - for example, some autistic individuals have lower cognitive empathy but heightened levels of affective empathy. The definition in this blogpost focuses entirely on cognitive empathy, but I bet the people telling them to be "more empathetic" were talking more about compassionate empathy.
[1] https://embrace-autism.com/the-different-types-of-empathy/