If you're looking for a textbook answer, I can't give you one.
I can however, give you my own, in accord with the parameters you laid down:
Life and the world are complex. The result of actions we take are not always immediately apparent. Sometimes we may seek to do good, and the end may be ill for some... other times we may seek ill and accidently benefit someone.
I'd say that being morally conscious in the sense you define it, is having an awareness of this complexity, and caring about it, and an appreciation that we do not always know where the results of our actions will land when we intervene in someone else's life. Such a person will try to consider the unintended consequences that might accrue to his actions, and how it may affect others who may not seem to be directly involved, or bear on future events that may come to pass.
Such a person would typically be slow to intervene and cautious to give advice, until they felt they understood the situation as completely as possible. Rather than the person who tosses some money at the problem to salve their conscience, this would be the person who gathers information and thinks things through carefully before committing to a course of action.
"Advice is perilous, for even the wise cannot know all ends." - Elrond