Political Memes

Sympathy is surely more about showing that you care about someone and their feelings.

Not an element of any definition I've seen. The phrase hidden sympathies exists.

As the rest of the Merriam-Webster entry states, it's a more general word:

Sympathy has been in use since the 16th century, and its greater age is reflected in its wider breadth of meanings, including “a feeling of loyalty” and “unity or harmony in action or effect.”

Empathy implies the general sense of sympathy defined before: an actively shared/vicarious feeling is a shared feeling.

Every word has some degree of vagueness. Some (like you) claim personally feeling as another feels isn't essential: it's more about understanding or caring. That empathy is about a serious effort to imagine or understand another's experience as if it were one's own, which sympathy doesn't imply.

While that sense is admissible, the word empathy without qualification doesn't communicate that sense exclusively, so I don't fault people interpreting feelings from it. Wouldn't following empathy (either sense) mean picking up the room for interpretation & communicating a distinction more clearly or perhaps jumping straight to the reasons driven at by empathy?