I've always taken Elizabeth's comment about "I'm not a great reader" to be countering the fact that the Bingley sisters and Mr. Hurst seem to be equating reading to being socially backward.

"Do you prefer reading to cards?" said he; "that is rather singular."

"Miss Eliza Bennet," said Miss Bingley, "despises cards. She is a great reader, and has no pleasure in anything else."

IMO, Elizabeth is not putting down her own ability to read or even denying that she likes to read, but reacting to their put-down of reading.

I've always thought of Mr. Darcy's addition of reading to his list of accomplishments to be a way of letting Elizabeth know that he values reading and those who take pleasure in it, even if his friends do not--almost like he was embarrassed that his friends would criticize someone for being fond of reading.

He may well be trying to set his sister apart from other women, but I think he is also trying to set himself apart from his friends and to indicate that he does not share Caroline's high opinion of herself.

I also think that with his comment of

"I cannot boast of knowing more than half a dozen, in the whole range of my acquaintance, that are really accomplished."--he means to specifically exclude Caroline Bingley. She is speaking of herself as though she would certainly be considered a woman of accomplishment. Her comment that "no one can be really esteemed accomplished who does not greatly surpass what is usually met with. A woman must have a thorough knowledge of music, singing, drawing, dancing, and the modern languages, to deserve the word; and besides all this, she must possess a certain something in her air and manner of walking, the tone of her voice, her address and expressions, or the word will be but half deserved"

His comment on the reading, to me, seems to imply 'Not so fast--anyone who puts down people who enjoy reading and values books only as a collection rather than for what they contain, does not fit my definition of "really accomplished" either.'