numerator = beats per measure
denominator = which note "gets the beat"
3/4 time has 3 beats per measure, where quarter (1/4) notes get the beat.
6/8 time has 6 beats per measure, where eighth notes get the beat.
etc...
That gets to the concept of simple and compound meter. In simple meter the the beat gets divided in two and in compound the beat gets divided in three. (Complex is a mix, like 7/8 may be divided 1-and 2-and 3-tee-ta).
Ultimately, the beat can be more interpretive. The 2nd movement of Beethoven’s 9th symphony is in (mostly) 3/4 but it is so fast, no conductor beats it in three, they beat in in one.
I've never heard this. I've only ever heard it as "1 2 3 4 5 6"
One thing about 6/8 that is not literally communicated by the notation is the rhythm: the first and fourth beats of the bar are stressed. But not equally, because that would be indistinguishable from 3/8.
And 6/8 says nothing about the beat structure within the 6 beats. You're stating the most common occurrence, but that's not the definition of 6/8.
The point stands that time signatures are not rational numbers (as you acknowledge, the unit of the denumerator can be scaled) so 3/4 and 6/8 aren't part of some equivalence class. They have different meanings which are only sustained by the presence of a cyclical pattern of stressed notes. So if there's some music theory you can link to which elaborates on all the possible different rhythmic patterns of 6/8 vs 3/4, that would provide examples of the distinction we both agree exists.