Q:

the meaning of the verb 'care'

Here's what Jane Goodall said in a movie trailer of 'War for the Planet of the Apes':

No one anointed us ruler of this kingdom. It’s not ours to own. We are simply one of the animal species on our planet, not separate, not superior. We hold in our hands a responsibility to preserve harmony in the natural world. To demonstrate compassion towards all who call this home. So many living creatures have feelings, know loss, suffer pain. We must understand the mutual dependence between humans and animals. Only if we understand can we care. Only if we care can all be saved.

In the last two sentences, the verb 'care' has been used.
What does 'care' in this context mean?
Which does it mean?

(1) to feel that something is important and worth worrying about
(2) to like or love somebody and worry about what happens to them

(These are definitions found in this dictionary.)
And to ask a more general question, are these definitions mutually exclusive when enough context is provided as here?

A:

The only definitive way to know the answer would be to ask Jane Goodall herself.  But I think we can understand it from analyzing her words.
To understand the specific intended meaning of care, I think the two most important sentences are:

To demonstrate compassion towards all who call this home. 
We must understand the mutual dependence between humans and animals.

Now for reference, let's pull up your two definitions.

(1) to feel that something is important and worth worrying about
(2) to like or love somebody and worry about what happens to them

Definition 2 implies a personal connection to an individual.  You love your dog, you love your mom, you love your best friend.  You cannot truly love a large group of animals you've never had any interaction with.  You can like the IDEA of them and appreciate them, but you cannot form a bond with someone/something you've never met.
Definition 1 talks about feeling that something is important and worth worrying about.  This could be a person, but it could also be a social cause, your grades, taxes, an inanimate object.  It's more open ended. 
I believe definition 1 is closer to the intended meaning.  If we are to "demonstrate compassion towards all who call this home" then that is a very large group of individuals.  I think arguably one could not love every single animal on the planet.  But one could certainly be generally concerned for their well-being as a group and think them worth worrying about.  
I think this is reinforced by her idea that we should "understand the mutual dependence between humans and animals."  She isn't calling for us to love every animal, but to be compassionate and concerned for animals that feel much the same way we do.  This is more reflective of the general "to feel that something is important and worth worrying about" as opposed to liking or loving an individual.
But, I will also say that with words like this, you cannot necessarily separate the two definitions.  To answer your question if they are mutually exclusive, I would say no.  A word like "care" will have many definitions.  And if you experience #2, you would certainly experience #1.  But you could experience #1 without loving a particular individual. Another dictionary would also provide a very different definition that could easily combine those two.  There is no simple, definitive definition for a word like "care."

