To me, most orgs don't practice any kind of sensible methodology at all. Waterfall implies a rigor they don't have.
Behind the scenes the PMs are making Gantt charts and deciding what's in and what's out and who is going to do it, without the team having any input or really getting to cost it out.
XP and agile generally was supposed to about shortening the communication and iteration gap between customer and maker. I almost never really see that.
I meant in my experience, the methodology most companies I've worked in seems to follow is capital-C Chaotic. Requirements change at the whim of customers or leadership week to week, pivots are frequent and unpredictable, leadership gets fired/quits and the whole plan changes, lots of effort is wasted on stuff that then gets ignored, and customers are sold one thing but delivered another.
I don't believe organizations or companies follow either Agile (any brand of it) or Waterfall. I don't think what they do can be called a methodology at all.
Then there's the definition - like, is SAFE really agile plus so many other hybrid approaches that have veneer of agile as long as we get all requirements up front and have detailed plan.
Actual waterfall looks very much like actual agile, in that it is designed around iteration loops. The primary difference is that waterfall prescribed steps in the iteration process, and agile is just a set of principles in a manifesto.
Edit: reference: https://beza1e1.tuxen.de/waterfall.html
TLDR: the original impetus for waterfall is basically what we call agile today.
Someone copy-pasted a random chart from a paper (one the paper specifically said was too problematic) into a DOD process spec, that turned into a standard because the DOD loves to standardize everything, and big companies all adopted the fundamentally flawed approach and called it waterfall.
This excellent writeup by David Olson gives both the history and the correct understanding; The Myth of the 'Waterfall' SDLC - http://www.bawiki.com/wiki/Waterfall.html
Most of what people are describing here would be unrecognizable to any government program office.
In that case, I don't believe companies follow either. I've never seen anything as principled (in any form) practiced anywhere.
Companies claiming to do Agile were usually doing some rituals and cosplaying as agile. I don't believe I've ever seen a company doing "waterfall" or anything resembling what your link describes either.
They mostly do chaos-driven development.
No, Waterfall was not agile, it was the diagram from Royce but not what he recommended from his paper (which tore down that diagram). What Royce added to that diagram (fundamentally, just common sense with feedback loops) was closer to agile, though. Royce himself never called anything Waterfall, but what was termed Waterfall was the bad process he tore apart.
