Care for, Care about

[Madness1]

I am having trouble understanding the difference between these two.
I know that "care about" means that something is important to me, but what about "care for"?
I've heard that "care for" simply means that I like something and we usually say we "don't care for" something, which is to say that we don't like it. Did I get it right?

I have a couple of examples. Can you tell me if they mean exactly what I just said they mean?

1. All he cares for is money. / All he cares about is money.
2. I don't care for dogs. / I don't care about dogs.
3. "Why haven't you eaten your vegetables?" - "I don't care for them." / "Why haven't you eaten your vegetables?" - "I don't care about them."

There is also an interesting example I have found online which I just can't get my head around. It just confuses me.
https://www.nytimes.com/2021/10/28/us/texas-nurse-death-penalty.html
“Even if that were true and that he somehow was trying to prolong their illness, for someone to do that — kill their patient and then try again,” he said, “you have to be the kind of person who has no empathy, who does not care for another person, who is unconcerned with their well being, who feels no guilt.”

What exactly does this mean in the example above? And how would the meaning change if it said "care about" instead?

 

[Tarheel]

Apparently, "care for" is used the way you describe. (I am used to the one the means "take care of".)

As for your examples, the first two are okay, but for the third say: " I just don't like vegetables." (Or something like that.)

In your last example I say "does not care about other people".