I don't see a major difference here, I mean, I could care about someone without them knowing, or actively care about him/her also without his/her knowledge.
Ever tried to fix your friends problem while making sure he has no idea that you fixed it?
ofcourse I only saw such an example in movies & Anime but still a possible one nonetheless.

I guess all of those depend on whether we are talking about the activity or the emotion, the prior requiring more than one individual, while the latter being a singular experience, or something.

well, we can pretty much find people who love, hate, admire ...etc another person without the later even knowing, so what's make caring any different?

Good points, you're right. I think the reason I identified the second example as not caring, despite having "thought" about it, is because "caring" might only qualify as caring if more than one person recognizes it as such. You can't really care about someone unless they exist, can you? It requires two people at the least, if we are on the subject of empathy.

What I mean is; is it caring if you think about the other person, but the other person does not know of it?

[Part 2]

Giving the previous inputs, I would think that both situations are actually caring (with objectivity), just think about someone -even if you do nothing- IS caring, not caring however would be something like: I see my friend crying -don't think about it- and immediately continuing with my current activity (rolling the joint).

srry if i bored you but that's what happens when you catch my interest x)

[Part 1]

Actually .... you're 50% correct regarding the examples, although you got the jest of the correct answer too,

	Well, if we define "caring" as actually doing something about the others ills

Exactly, it's how you 'define' caring, to me, I think caring about something is to think about it for more than 10 minuets, I think that thoughts are alot more valuable than actions on many levels, a tiny example for that: You help 'A' because you care for 'B' (& 'B' cares for 'A'), if we define caring by the 'actions' then one would say that you care for 'A', which is not the case at all.

	there can be situations in which you don't do anything about it exactly because you care about the other person, can't there?

See? you arrived at the same conclusion, if we consider actions one'd say that you didn't care, but a whole different story if you considered thoughts.

Well, if we define "caring" as actually doing something about the others ills, then the first is objectivity, and the second is not caring at all. But surely, there can be situations in which you don't do anything about it exactly because you care about the other person, can't there? Sometimes, they need to fight their own battles

I get what you're saying, though.

mmmmmmm, you caught my interest there.
About the Blurry line between objectivity and 'not caring', don't think it's really that blurry, it's actually quiet easy to differ them with the right inputs ...
Lets make a game of it:

Situation A:
I walk by my crying friend, I stop and ask her "What's wrong?", she says that she had a fight with her BF and the bastard hasn't called her since then, and I reply "well, seems to me that you haven't called him since that too, doesn't that make you in the same wrong?"

Situation B:
I walk by my crying friend, I look at her and I already know it's gonna be some stupid trivial problem that will make me smash her in the head 30-40 times if I hear it, plus she doesn't listen to any one and probably will forget our conversation 2 seconds after it's done, so why bother? ... and I carry on with rolling the joint ...

So, which situation is an example of objectivity/not caring ??

I think a person with sufficient self-knowledge, control over their emotions and a rational mindset will find it hard to be strongly emotionally affected by problems of other people, even friends. That does not mean they do not care, though. It just means they can detach themselves and look at it from an objective point of view. Although the line between objectivity and not giving a shit is quite blurry considering traditional morality, and how people tend to view such individuals as "cold," so I'm not entirely sure. All I know is that if you can keep a cool head in times of despair, people will appreciate it as long as you stay considerate, which is a choice, and not a reaction.

I guess you're right, I mean I did wanted to know what was going on, but I think i it was curiosity more than anything else, you know, the kind of curiosity you have when you see an accident from afar and you wanna go take a look at the victims ...