This phrase also has two significant beats. There's just a barline in the middle. See my point?
I would argue that We Are The Champions is actually a 12/8 groove (conceptually triplets in 4/4). And we would both be right!
The convention is that these two time signatures stand-in for two different actual perceptually different music feelings.
The "tactus" (the place we feel the beat) certainly can be subjectively moved to different levels in the hierarchy. Furthermore, you're right that the bottom number is basically just a notation preference (6/8 and 6/16 are effectively identical, though nobody uses the latter).
But the whole point of anyone describing the difference between 6/8 and 3/4 is that they are using the convention to describe the actual feel difference between dividing the same amount of time into 2 vs 3, the hemiola issue.
I basically was taking issue with seeing a reply to a post that was highlighting this difference that is a real perceptual one (not just a notational one, but described, unfortunately, with focus on notation) with the claim "there is no difference".
I apologize for the aspects of my reply that were ad hominem instead of just critiquing the post itself.
It is impossible to quantify or intellectually stratify feelings and perceptions. All I have to say is "I feel it differently" and now your stratification is incomplete.
But you can say without a doubt that 3÷4 === 6÷8.
Of course I am aware of the conventional approach to playing 6/8 that results in this perception that beats 1 and 4 are not of exactly equal weight, and of all the different ways you can play two against three, three against four and so on. Polyrhythm doesn't stop at hemiola by the way. Are you also feeling five against three, seven against five, etc.?
I can show you examples of music in 6 where you may be unable to find the downbeat at all. So you can't say for certain how I or anybody else is going to hear it in every case. These prejudicial approaches to music cause closed ears.
The whole point is that when we communicate, we use (often mediocre) conventions.
3/4 and 6/8 are notational things with nothing but convention determining their meaning. But same with the entire notation system, it's just shapes with no meaning until we assign meaning to it.
The only really valid theory of music is one that embraces the subjective psychology of it (music cognition etc).
To say that music where one can't find the beat is "in 6" is already weird. It's only "in 6" if there's something about a 6-beat meter in the mind of some person, listener or performer. We can listen to something together and you can experience it in 6 while I experience it in 7, if we're capable of those different subjective interpretations (and the sound content will affect whether that's easier or harder).
The whole point is that while the prejudicial approaches to music are wrong (music is 100% subjective), the purpose of the original article (by my charitable interpretation) is to describe two different subjective experiences that are possible, hemiola being about the most basic introduction to such concepts for beginners.
So, I took your original "no difference" comment to be a denial of the existence of two distinct subjective experiences just because the notation can be seen (ignoring convention) to have no indication of the difference.
You can see how I could take your post that way. I think it's really valuable for anyone learning about music to understand these subjective experiences and their differences. We agree that it's crappy to assert that these subjective experiences are in the notation, but I was willing to see "6/8 vs 3/4" as a communication stand-in for the subjective difference that is real.
Surely, some people say "6/8 vs 3/4" and mean to assert some difference that is just not really there. And others use it to refer to the hemiola distinction of subjectively parsing an accent in these two ways. The former is a delusion, the latter is a valid insight.
P.S. In case you haven't seen it, one of the neatest rhythm things out there is http://bouncemetronome.com/ — it goes intro demostrating beautifully any level of polyrhythms, including offset ones evolving ones, rhythms that phase over time because of differently changing tempos, and a ludicriously long list of other things… a fun crazy tool to get anyone into stuff way beyond whatever conventional music system they may have learned otherwise. I doubt much of it would be entirely novel to you, but you may still appreciate it. The options are enormous, hard to find the limits.
If your phrases really go ONE two three FOUR five six then notate your piece as 3/8. 6/8 is for ONE two three Four five six SEVEN eight nine Ten eleven twelve, where beat four is significant, but not as significant as beat one.
Of course you do have patterns of bar phrases, not all first-beats-of-the-bar are equal in a larger piece, and where you draw the line between a bar and a phrase is ultimately a matter of taste and judgement. But a player will absolutely understand 6/8 and 3/4 differently; a run of quavers in 6/8 is ONE two three Four five six, a run of quavers in 3/4 is ONE two Three four Five six.
Of course you can take 3/4 music that happens to use a lot of two-bar structures and play it with the same feel as the same thing notated as 6/8.
But the point of comparing the two is not to highlight that case where the same music could be written either way. The point is to describe how if you take the same 6 8th-notes and give them a 3×2 feel, it's musically different from a 2×3 feel.