Secondary offering defined -- << A secondary offering (or part of an offering) is an offering of shares by existing shareholders (often insiders) rather than by the company itself. Sales of shares by the company itself—whether a PIPE, an underwritten offering, or a registered direct—are primary rather than secondary offerings.>>
There seem to be two definitions. Many people -- see investopedia and wikopedia and, I can tell you, at least some Wall Street corporate lawyers, use "secondary" offering to mean any offering after the "initial" public offering. Others use it as you say it should be used. When I hear the phrase used in my practice, I generally ask the corporate lawyer for clarification, e.g., I will say, do you mean, secondary, as opposed to a primary offering. A primary offering clearly means an offering by the company of its own shares. And frequently, they just mean, an offering other than an IPO.