Book publicity is one of several related and sometimes confusing or nearly synonymous “p” terms.
Someone does promotion (which often includes public relations) to achieve publicity.
Publicity is lots of people knowing about your book and hopefully buying copies and/or urging others to buy.
Promotion is all of the efforts intended to achieve publicity. Although publicity is the end result of promotion, many people call themselves book publicists and relatively few call themselves book promoters. (Publicists used to be called "press agents"). A publicist or promoter can guarantee to provide promotion, or public relations, but cannot guarantee that you or your book will achieve publicity.
Despite its name, public relations is not directly concerned with relations with the public. Media are intermediaries. Writers hope to attract the attention of media people by sending out press releases, or by contacting journalists, editors, bloggers, talk show hosts, TV producers and movie makers.
Promotion includes more than public relations. It may include public appearances, publicity stunts and platform building.