The Royce paper is actually still worth reading. It's well written and if you can step over the 50+ years of history since then also still somewhat informative. And of course the word waterfall doesn't actually appear in it. That word was slapped on it by others afterwards. And of course Royce vaguely suggests that going through the steps more than once might be helpful. What we now call iterating.

The key concept for what we now call agile (which mostly boils down to variations of scrum these days) actually has roots dating back about half a century. People were suggestion spiral development at some point (converging on some final thing) as early as the eighties. There was the whole 4+1 model which got caught up in the whole UML movement and it's associated process.

The key innovation that extreme programming (one of original agile methodologies) brought was that increasing the iteration frequency and giving up on the notion that you can do a lot of requirements and design up front. Things change. Every iteration. So having a lot of requirements and design documentation to update every cycle is counter productive. Dropping that was considered extreme. But then people realized that that worked just fine and that these designs were kind of low value artifacts to begin with. Not really worth the hassle.