The American Heritage Dictionary[1][2] and the Merriam-Webster Dictionary [3][4] are two that come to mind, but I'm sure that you can find others.
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.
Understanding other’s feelings seems to be only part you’re focusing on, but it’s not enough to have empathy. The dictionaries you’ve linked to have each a list of skills and I don’t think each stands on its own. It’s a combinations of those.
I’m sure AI already have, and cenrtainly will have it perfected in the future, an ability to “understand” feelings and emotions if I use the term “understanding” the way you do with sociopaths - the ability to analyze and act on. Certainly it can not be called empathetic.