This has been so endlessly repeated without thought that people assume it is a fact when the Truth is far from it.

Excerpts from wikipedia - https://en.wikipedia.org/wiki/Waterfall_model ;

* In 1983 the paper was republished with a foreword by Benington explaining that the phases were on purpose organised according to the specialisation of tasks, and pointing out that the process was not in fact performed in a strict top-down fashion, but depended on a prototype.

* Royce's five additional steps (which included writing complete documentation at various stages of development) never took mainstream hold, but his diagram of what he considered a flawed process became the starting point when describing a "waterfall" approach.

* In 1985, the United States Department of Defense captured this approach in DOD-STD-2167A[citation needed], their standards for working with software development contractors, which stated that "the contractor shall implement a software development cycle that includes the following six phases: Software Requirement Analysis, Preliminary Design, Detailed Design, Coding and Unit Testing, Integration, and Testing.

The "Waterfall" is what is imagined as a "Ideal and Rational" process (see D.L.Parnas' paper A Rational Design Process: How and Why to Fake it for motivation - https://ieeexplore.ieee.org/document/6312940). The stages in the model and the issues to be considered in each stage are what is important NOT their order. In practice it was always a iterative spiral model described by Barry Boehm - https://en.wikipedia.org/wiki/Spiral_model