This images should hopefully clear it up a bit, i may have referred to the wrong scale notes (eg it’s G root for mixolydian)
Some scales have the same notes, however they are played in a different order; think about A minor and C major. Both the same notes, just one starts at C, the other at A. A minor sounds darker than C major, even though they use the same notes. It’s to do with how the accidentals are spread through a 12 note scale. You may be playing the same notes, but the gaps between notes are different, giving different tension/release.
EDIT: Also, you’re correct, it could be referred to as “binary rotation”, although I think the “classic” way of referring to it is usually a “church mode” on piano at least. Unfortunately I’m not a guitarist, so I’m unsure on how easy it is to “rotate scale” in comparison to a piano as your fingering will likely be totally different. I often get tangled chatting to my guitarist buddy about these things.
The website you linked is fascinating by the way, I’ve been down that rabbit hole once before, good fun if a bit mind melt-y. Hope this clears things up, gimme a shout if this still needs a bit of clarification.