The former is, indeed, Fi in my own experience. Fe doesn't need to internally validate the feelings of others and relates to others pain and joy without drawing from internal experiences. As a Fi-dom, I relate to others by drawing on my own experiences and "recreating" their feelings inside me, so Fi is more tempered when it relates others I think. Both Fe and Fi possess empathy but it is sourced differently.Which feeling function can recreate the feelings of others internally through imagining and then act based on that feeling?
To answer your last question, my INFJ husband and an ENFJ I know do act as though they feel the pain of others, as close as one can without being the actual person they are relating to.