Consider:
- I have a habit of biting my nails when I’m nervous.
- I have the habit of biting my nails when I’m nervous.
The key in both examples lies in the intent of the sentence. Clearly, the only difference is the choice of article. The rules of grammar that govern the use of articles can be not only prescriptive, but also descriptive. The choice of article, in some circumstances, can be what determines the meaning of the sentence.
In example 1, there is a collection of habits in which one (a) is biting my nails. We can use our understanding of the use of articles to determine that the narrator’s intent is to describe a habit among many habits (either his own or a more general collection of habits).
In example 2, the specific habit of biting one’s nails is a habit that exists among a collection of people who have this habit of which I am one. The choice of the definitive article is used to define the entire phrase (habit of biting my nails) as specifically important. In this context, the particular habit is one that exists among many people with this habit, not among many habits.
In the context of an example in the question, it’s not quite as neat, but:
She was given the responsibility of taking care of her sister.
- She was given a responsibility to take care of her sister.
- She was given a responsibility to clean her room.