It makes no sense to me that it be Lydia. Besides, Jane says,

"[...] I am sorry to say that by his account as well as his sister's, Mr. Wickham is by no means a respectable young man[...]"

so, she has been given answers by Caroline.

I can see Jane not mentioning that Wickham's father was a steward - to Jane, bringing up someone's lower beginnings would be inexcusable unless there was a reason for it. She might NOT have said that Elizabeth liked Wickham for Caroline to assume so. If she said that Wickham had told Elizabeth his story and Elizabeth had told Jane, that would be enough. I rather thought the 'tone' was one of condescension and superiority.