I don't agree with the notion that you need to "understand" to be empathetic. If a friend tells me that they're going through a rough time I feel like it's disingenuous for me to tell them that I understand their situation, even if I myself have been through difficult periods. I can't and don't feel like I should give the impression I can understand the nuance of their situation. I've found it's much more productive to be supportive in situations like that by saying things like, "I know I can't understand fully what you're going through right now, but that sounds really hard and if I can help in any way please let me know" versus "Yeah, I understand what you're dealing with."
If empathy is putting yourself in another person's shoes and that's the understood general definition... I don't think it's genuine to say that, "Yes, in these few seconds/minutes/hours since you've told me your problem I have gone through the effort of visualizing life from your perspective and I now understand all of your issues." Even if "I understand" is communicated with good intentions I feel like it rarely looks to productive discussion.
Well, right, you might not immediately understand and identify with someone else's problem. That is, you won't empathize with them. But you could say that you sympathize with them, if their pain causes you to feel pain as well (or joy, or...whatever). Or their situation may cause you to feel compassionate, if their suffering makes you want to help. I think that there are already words that name the feelings you're talking about.
"Empathy" has a generally-accepted meaning. You can disagree with it if you'd like...but it will hinder communication to use words in ways that they aren't commonly used.
Well, empathy is a word with a definition, not a directive as to what is helpful to do.
Empathy is the ability to replay another person's state in your mind so that you can understand it. It's not directly connected to what you do with that information, though. You can just as much use empathy to manipulate someone since it lets you model mental states well.
In your particular example, if what you say is true, having empathy would mean understanding that the person probably wouldn't care for you to tell them that you understand. :P
If empathy is putting yourself in another person's shoes and that's the understood general definition... I don't think it's genuine to say that, "Yes, in these few seconds/minutes/hours since you've told me your problem I have gone through the effort of visualizing life from your perspective and I now understand all of your issues." Even if "I understand" is communicated with good intentions I feel like it rarely looks to productive discussion.