Hi again, Loco Mo; I'm going to take a stab at answering this before I log off for awhile.
Definitions from the dictionary are, in my opinion, not terribly helpful, so I'll offer another example, which I hope will help. And you are correct: Pity is a rather pejorative term.
Sympathy involves an element of empathy, whereas pity does not. That may sound confusing, but I'll wager you and the other forum members know what I mean. For example, I can feel sympathy toward native North Dakotans who want to travel to different states and countries but do not have the financial means to do so, but I pity people here who have no desire to leave this region and broaden their cultural horizons.
Does this make sense? If not, I'm sure someone else can more clearly elucidate the variations among empathy, sympathy, and pity.
Also, you are correct in that a lot of people prefer the term empathy, even when it is not technically correct. Empathy requires the capability to identify with what someone has experienced or is experiencing.