LT for me those 2 statements mean something different.
If I am at work and my boss says “I care about your success” I would take that to mean she cares very much that I do my job right, push forward and continue to make money for the company. I would not internalize that statement as meaning she cares about me. Don’t get me wrong, she would be sad if something bad happened to me so on a level she does care.
If my best friend said that same statement then I would internalize that she does care about me very much. I think it depends on the relationship. Coming from a therapist- I wonder if I would think that statement had a bit of selfishness to it? “I want to see you succeed because then I know I did my job well” but..pay no mind to me, LT-I’m stubborn so I would probably tell a therapist they are selfish for saying that because they would want to know they did their job to help a client.