As usual m-w is much sloppier when it comes to making distinctions between similar words, due to their philosophy that common misuse redefines words[5], but even they make a pretty clear distinction between empathy and sympathy and it's clear that sympathy is the one where one actually emotes with the subject.
[1] https://www.ahdictionary.com/word/search.html?q=empathy
[2] https://www.ahdictionary.com/word/search.html?q=sympathy
[3] https://www.merriam-webster.com/dictionary/empathy
[4] https://www.merriam-webster.com/dictionary/sympathy
[5] And they aren't even entirely wrong, as anyone can literally tell you.
However I'd think some sociopaths can be good at logically cognitive intelligence wise understanding how a different person feels.
But being unable to actually feel real empathy, I wonder if they might mistake their logical thinking for being empathy? Which they will instead never know what it is