You've really already got it!
The way I explain it is (and this is only a cut and paste of what is essentially a paraphrase of what you wrote)...
Being empathic (or empathetic) means a personal identification with the feelings, thoughts, or attitudes of others. Tell me about what you are going through and I can empathize if I have gone through the same, or very similar, life experience.
Being an empath, and especially being an impaired empath, means taking on the feelings, thoughts, or attitudes of others as if they were ones own even before there is any conscious awareness of what is happening and regardless of whether I have gone through the same or similar life experience or not.