Webster 1913's definition is about four decades out of date. As Webster alludes to, the original meaning of the word "publicist" was a writer on public law, from the French publiciste, via the Late Classical Latin publicista, "an expert on public law," which was originally derived from Latin ius publicum, "public law."
However, the modern meaning of the word publicist, which dates to around the year 1930, is a person whose profession is to generate publicity for and manage the public image of an individual person, corporation, or product.
Among other tasks, a publicist may be called upon to write press releases, arrange for interviews or favorable press coverage, coach clients on how to behave in public appearances, manage or advise marketing campaigns (whether viral or traditional), and even design and/or carry out publicity stunts.
A successful publicist must draw upon a variety of assets including outstanding oral and written communication skills, excellent interpersonal skills, a network of contacts in the media, quick thinking and crisis management abilities, and imagination and creativity.