There is a huge difference between caring about someone & caring about what they think.
When you really care about someone, you care for them, without being judgemental (& vice versa ideally.)
However when you care about what someone thinks about you, you don't really care for them as much as you care about their judgement of you. That how do they judge you.
I don't care what the world or my extended family thinks about me or what I am doing, but I still care about my extended family. If they are ill at 2 A.M. I will rush to get medicine for them, not because I would look good to them, but because that is the right thing to do.
& that is a subtle difference that people really need to understand.