You would be correct.
Or simpler terms-
Sympathy:
I'm so sorry you're in pain.
Empathy:
I feel your pain.
I am a little unsure about some of these dichotomies though, like the last time I read a book which seemed to support them, one which was campaigning for compassion and against empathy, it seemed to be trying hard to rehabilitate the version of sympathy in Adam Smith's book on the topic as contrary modern understandings of empathy.
There is a ton of publishing from Smith's time which in interesting ways prefigures modern writing on it, his contention that imagination and the ability to imagine oneself in the position of others more or less fortunate than yourself and then later to consider all matters as an impartial, invisible other (which Rawls' later writing would describe as the veil of ignorance or hypothetical first position of anyone) all sound more like empathy to me than sympathy but I'm not sure that Smith would have had the same vocabulary.
It was all literary back then too, or moral philosophy, and the vocabulary was different reason, feeling, passions etc. and speculations about human nature at the heart of it all, only very slowly shedding the Negan like thinking of Hobbes. Hume's books asking if man was passionate or rational for instance, its philosophy not psychology but its not far removed from some pop psychology in what its searching out and better than most of it, there's literary works like "The Man of Feeling", considering people who tried to apply the thinking which I sort of approached reading with a "What the f**k is this now?" (as I do a lot of books, more than people realise).
Rousseau and, to a lesser extent Voltaire, were supposed to be the continential alternative to the anglo-saxon outlook, at least Rousseau really was, I read a US author, woman who wrote With Charity To None, a fond look at misanthropy, compare Rousseau to a sort of pre-hippy era "flower child" and I dont think she was wrong, although, again, the vocabulary was so different and you would need to do a bit of a work translation with historical context in mind to know it.
Then you realise that Freud, Jung et al were trying to do the same sort of thinking and using different terminology, some of it was defined better than others, some of it wasnt (did you know that prior to Freud's tripartite version of the unconscious Erasmus had one when he was writing about ideas like the Christian Knight? Language is different, obviously its not the same thing exactly, but there's something perennial they were aiming at) but a lot of this modern definition and dichotomy happens and they arent even aware of the precursory work.
