The person responsible for the strategic and operational oversight of an organization's entire digital presence, not just its website. Strategic Web Management Ch. 1 (1.2)
The word doing the work is entire. A web manager oversees the whole web stack: the domain names, hosting, business email, social profiles, analytics, and paid web apps, as well as the website itself. The job comes down to three duties, in order: reducing risk, maintaining continuity through staff and vendor changes, and improving the digital presence deliberately. Writing code, designing graphics, and fixing laptops are not part of the role; those are crafts a web manager rents and directs. The role is usually held by someone with another job title, such as office manager, marketing coordinator, founder, or volunteer.
For the full description of the role, its boundaries, how it differs from a developer or webmaster, and the skills that matter, read, What Does a Web Manager Actually Do? and download the free Web Manager Role Blueprint.