I must say I liked this definition. I definitely agree. If I may though, I would like to expand on this definition.
In a general sense, it seems to me that the definition to care about something is to have an interest in some respect related to the object of consideration. However we should differentiate the type of caring we are talking about here. Do we want to focus our discussion on the type of positive interest we would use the term care in to say "I care about my friend" or, "I care about my piece of pizza", or do we want to use the general sense of interest for then we must include the kind of caring that is also negative. An example of this might be when a person "cares" what a bully says for it is true that the bullied person does care as it does bother him (specific to the case of one who is annoyed which is certainly common).
I will go ahead and make a distinction to say that we are talking about care in the sense of positive interest in something. Now that we have the definition of care, we can discuss its application or to say "care for" something. It would seem to be that there are many respects in which a person can care for a single object (currently restricting to inanimate object for simplicity). Lets take the example of a vase. We could care for the vase because of its immediate appearance (drawing on its side or its specific shape). We could also care about the vase for association, for example it could have been your great grandmothers vase. We could care about the vase for the purpose for which it serves in the sense of extending the life of flowers that have been cut and brought inside and therefore brightening our room. We could perhaps say we care about it by association with our self if say we have owned the vase for 30 years (arbitrary number) and we associate it as a piece of who we are. However an argument could be made that this is an extension of the association with something else point.
These examples imply three ways to care for this vase. Its immediate appearance, its association towards something else, or its practical purpose.
Now lets try to extend our consideration towards a human. We could still have the same reasons to care for a person as we did the vase, however we must add some new types of caring. To determine what possible types of caring are added to the person, lets first consider what's different about the person from the vase. Well first the most obvious, that the person is alive. Another is that with a person we must accept this idea of personal identity in some sense. Another still would be that now we are considering the thing being cared of resembles the thing caring for.
From these points we can see that we can care for a person for the basic fact of being a life, we can care for something based on who they are as a person and even what they have done in the past (however a point could be made that what they have done in the past is what makes them who they are as a person), and we could also care for the person for being like us. However even these points seem to sound like the general points with the vase. Caring about a person for who they are is like caring for something for what it is of itself at that moment, caring for a person for things in the past is like caring by association, and even caring for being like us is a type of association. A point could also made that caring can also result from certain social dynamics in several unique ways, however I would say these are still like the caring for what it is like in of itself and caring for what it is associated with.
So it seems to me that the main ways of caring for something are:
Caring for what it is
Caring for what it represents/is associated with
Caring for its practical purpose
As for the question of do you have to care about every part of something, that seems to me to be unnecessary because things have many different parts. For example you could care about the vase because it was your great grandmothers, but you could still think the thing is as ugly as could be. At this point though we should differentiate the way we care about something. With the vase we would say that "we care about the vase because it was my great grandmothers". You could also care about something in several respects and these respects can be different from someone else's. For example in the case of you caring for the vase because it was your grandmothers, a friend who didn't know it was your grandmothers vase and thought that the drawing was actually good might care for the vase because of they thought it was beautiful, and maybe also because of its ability to extend the life of flowers indoors (lets assume you have allergies so you also did not care for it for its ability to hold flowers).
By this logic, care is highly situational and subjective. Now we can talk about the different ways we care about the vase to say we create a set representing the different ways we care about something. In the case of your movie, there are a set of ways your friend and you both cared for the movie (assuming you liked the movie), but its also true that each of your sets were different. In your case your set included that specific point you pointed out. As for a discussion of who cared more or less, well that's another discussion entirely.
Well that's my two cents. Hopefully that was helpful