[1] "the ability to understand and share the feelings of another"
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.
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
"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.