According to the definitions, my take is there's little difference between sympathy and compassion.

I think the root emotions branch towards others such as forgiveness and gratefullness, resentment and anger. It's we who decide to take that step or not. As children, we probably weren't aware of all the choices.

It seems the difference between sympathy / compassion and empathy is the matter of degree which we give ourselves to it. Either one can lead to action or inaction. To sympathize or empathize may or may not include an offer of assistance.

Not assisting does not imply apathy. Apathy would be a total avoidance of any emotion and would not include assistance in any measure.

I think we can say from reading your posts that apathy is not a concern for you. As well, symapthy and compassion are obvious.

For me, empathy is a tool. Why use a 20 pound hammer when 10 pounds will do?

We can tell if empathy or sympathy is more appropriate by guaging feedback. In either case, it's important to avoid co-dependancy. I think that might be the line you've tried to define.

"Our greatest glory is not in never falling, but rising every time we fall." Confucius