Here's a contemporary example, from the Journals of Lewis and Clark: "At 11 A.M. we met a Captain McClellin... of the U States Army assending in a large boat. this gentlemen an acquaintance of my friend Capt. Lewis was Somewhat astonished to see us return and appeared rejoiced to meet us. we found him a man of information and from whome we received a partial account of the political State of our country, we were makeing enquiries and exchangeing answers [etc.] untill near mid night"

In Ch. 4 of Emma, Emma expresses doubt whether Robert Martin is a "man of information" outside of his own line of business. Harriet responds that in addition to reading the Agricultural Reports, he reads to his family from the Elegant Extracts and has read the Vicar of Wakefield. (Iirc, the AR's were a periodic publication of the Board of Agriculture, publicizing the latest farming techniques. That Martin reads them is evidence that he is no traditionalist clodhopper but takes an educated, up to date interest in his work).

In Ch. 8, Knightley in turn describes Harriet as "a girl of no information" who "has been taught nothing useful," has had no experience of the world, and wouldn't be smart enough to benefit from experience anyway.

So the context in which Emma is using "man of information" about both Martin and Churchill (Ch. 20) is of a man possessing some degree of culture. Knightley uses it to mean a woman with practical sense and judgment suitable to her station. Elizabeth uses it to mean that Darcy would have the knowledge of the world helpful in cleaning up the mess of Lydia's elopement. Clarke uses it to mean a man who was well informed about what he and Lewis wanted to know about current events.

JA also uses "information" not only as a noun denoting data, but as a verbal noun, denoting the process of informing one's self. "As to [Wickham's] real character, had information been in her power, she had never felt a wish of enquiring."