The mistake I believe you're making is in assuming that because most people reference Royce's paper for a description of Waterfall, that means that the definition of Waterfall is whatever the main subject of that paper is. This does not follow.
The paper describes a number (more than 2) of possible models. The main subject of the paper is probably the "2-pass waterfall" diagram in Figure 7. However, when people refer to this paper in the context of the Waterfall model, they mean Figure 2. The Waterfall model is called the Waterfall model because Figure 2 looks like a waterfall, with the water never flowing uphill.
Figure 7, however, was largely ignored. (Though Brooks later recapitulated it as "Build one to throw away; you will, anyhow". I don't know whether it was influential in the development of the Spiral model or not, though it seems like a logical progression.) As far as I know this model never even got a name attached to it. Its influence pales in comparison to Figure 2, which was the first and best published reference to a design process that almost everybody accepted as "ideal" at the time.
Note that when Brooks tells the story in The Mythical Man-Month about making the mistake of getting a large group of mediocre engineers to write specifications instead of giving the job to a small group of elite engineers because he didn't want the larger group just sitting on their thumbs for a year, this was all happening in the mid 60s. Here's another contemporary quote:
The most deadly thing in software is the concept, which almost universally seems to be followed, that you are going to specify what you are going to do, and then do it. And that is where most of our troubles come from. The projects that are called successful, have met their specifications. But those specifications were based upon the designers’ ignorance before they started the job. —Doug Ross, 1968
Winston Royce did not invent the Waterfall model in 1970, he was just describing the already-dominant paradigm as a prelude to proposing something different. The model had been around forever, Royce provided a convenient diagram (Figure 2!) in the course of trying to critique (or even replace) it, and the name "Waterfall" got attached later.
> Note that, in the section referencing DoD-Std-2167, the author of the DoD standard does state explicitly that he understood Waterfall to be one-pass.
Of course he did, and he understood it correctly. The thing he didn't understand was that it was a terrible idea.
Had he (properly) read Royce's paper, he would have seen that it actually advocated a slightly better (but still pretty terrible) idea, but that idea was not and is not the Waterfall model. The Waterfall model is the one in Figure 2 that looks like a waterfall. Hence the name.
> My own interpretation is contradicted further down in my comment by someone from that era (see the Larman paper linked). C'est la vie, I'm sticking with my interpretation.
Um, OK then.
Thanks for the link though, it was really interesting.