em·pa·thy
1. the intellectual identification with or vicarious experiencing of the feelings, thoughts, or attitudes of another.
care
1. a state of mind in which one is troubled; worry, anxiety, or concern.
I can identify and understand with others feelings,experiences, and attitudes, but I am not worried or concerned about what anyone else is doing.
1. the intellectual identification with or vicarious experiencing of the feelings, thoughts, or attitudes of another.
care
1. a state of mind in which one is troubled; worry, anxiety, or concern.
I can identify and understand with others feelings,experiences, and attitudes, but I am not worried or concerned about what anyone else is doing.
The context of the definition is equally important to the understanding of the definition.
You left out the associated sentences with your definitions.
a state of mind in which one is troubled; worry, anxiety, or concern: He was never free from care.
After reading the context it is clear that definition does not apply to the context in which you previously used "care".
When taking the context of the word into consideration, the definition that you should have posted and which fits the context of your previous usage was
to be concerned or solicitous; have thought or regard.
Feel concern or interest; attach importance to something
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they don't care about human life
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I don't care what she says
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they don't care about human life
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I don't care what she says
The second definitions being a much more accurate representation of how you used the word "care" and also necessitating a relationship to empathy.