For example on https://www.simple.com/, the logo in the header is simpler that the logo on the credit card.
They also show the 3 versions they use in a blogpost: https://www.simple.com/blog/simple-branding
[1] http://www.underconsideration.com/brandnew/archives/simple_b...
The fact that things rearrange dynamically as you resize your window on a desktop box is not the main target, the target is a design working equally well on your desktop, your table in both landscape and portrait, your much smaller phone, and so forth, so the designer doesn't need to maintain multiple versions of the site/page/app.
This is the original article which named the technique: http://alistapart.com/article/responsive-web-design/
Although the technique was being used earlier than May 25, 2010.
Adaptive Web Design (responsive but on server side, not client) is the other technique popular with websites aiming to support 96% of browsers well. Because it's impossible to do that nicely for mobile browsers.
Mobile first, offline first are the other two related terms you've likely heard. Meaning design for mobile first, or offline first.
Not just change, but remove flourishes and not entirely necessary elements.
> I've been seeing it used an awful lot in web design as of the past few years, often rather vaguely as some sort of feature.
It's a feature for cross-device compatibility, the point being to alter visible content to fit multiple device sizes.
It would be much more impressive to see the same parts of logos reused on different variants of logos - "The Man With A Gun’s Method" that is covered in the same article by Ilya Pukhalski.
For comparison: "The Hobo’s Method" - http://responsivelogos.co.uk/images/logo5.svg and "The Man With A Gun’s Method" - http://pukhalski.com/responsive-icons/responsive2.svg
Edit:
It is strange that author publishes (source: http://www.joeharrison.co.uk/projects/responsiveicons ) the information from Smashing Magazine containing the methods of responsive SVG and still uses the poorest method.
I think you've got it backwards. Joe Harrison first published responsiveicons.co.uk and the Smashing Magazine article is a later improvement on it. He didn't use the newer method because it wasn't around when he created the page.
In my view, "The Hobo's Method" is superior in that in keeps in-tact the core identity for each brand. Also, it helps to see real world examples.
There's a little bit about the history of the logo here: http://www.coca-cola.co.uk/125/history-of-coca-cola-logo.htm...
Slightly surprised that the "Coke" text from the largest logo wasn't used as the smallest.
(I guess there is a reason in the first place to display the information. Simply leaving it out feels very wrong. Or it was already unnecessary.)