Neither, actually. What you are asking about is *perception*. Most of the "Fe" answers are coming from NFJ s who are using Ni to gain insight on another's emotions through applying personalized pattern recognition, and most of the "Fi" answers are coming from NFP s who are using Si applying personalized past experiences.
Hold on.
The function that is dealing with other person's feelings is...well, Feeling.
It might get help from other functions to imagine someone else's state or be in that other person's shoes but "insight", as you state is not required to recognize someone else's reactions or feelings.
Of the two, Fi builds a framework, so it may know where the other is coming from without the need to see it played out in front of him.
I saw a lot of ESFPs who knew what the others were thinking without getting all the clues, just from the behaviour they could guess. (and they guessed good)
Fe is more concerned about feelings that are obvious. If someone makes an angry face, they will know that person is angry. But until it gets obvious they may not have a clue. (of course it can happen with inferior Fi too)
Fe is the equivalent of Te and Fi is the equivalent of Ti. As much as a Ti person is more able to dig down deep and spot the inaccuracies of a system or theory or whatever, a Fi is more able to see the person in depth and see through his personality.