Having an understanding of the feelings of another organism and being motivated to improve their state from one that they don't like (even if you don't act on the motivation, you feel it). That seems consistent with colloquial usage.

If you can use language to communicate with the putative empathy-feeler, you can ask them about it. They say things like "I feel sorry for so-and-so". (They might be lying, though.)

If you can't use language to communicate with the putative empathy-feeler, either because they only speak languages that you don't know, or because they're not linguistic creatures (e.g. are rats), then you observe behavior, looking for things like flexible invocation of supportive behavior when the organism needing empathy has negative behavior. (It might be a reflex, though.)

If you're not going to assume "everyone lies all the time about empathy" is the parsimonious explanation, you also shouldn't assume "everything is a model-free, emotion-free reflex" all the time, either. (Note that rats, like us, have mirror neurons: neurons that respond to pain both in self and in other rats.)