I would argue that the average person refers to a mix of both when they use the word "empathy":
caring about the other person's pain
being there for them
trying to see things from their perspective
wanting the other person to be better and wondering how one could help.
Classifying one as "a terrible problem" and the other as acceptable seems (at least) a bit pedantic. Specially when it comes to language, a dynamic phenomena in which words mean what (the majority of) people deem them to mean.
That's Big Mouth's Jay shit, right there