Instead, I find it useful to think about combinations and layers of twos and threes. So the different being talked about in this article is about whether the beat pattern is more of three groups of two, or two groups of three. Convention has mapped those, typically, to 3/4 and 6/8 respectively, but again that's entirely arbitrary.
Likewise 9/8 time is usually three groups of three. 4/4 is two groups of two (though the second two often have slightly less emphasis, whereas 2/4 time is just single groupings of two). 7/8 is a group of three and two groups of two (could be 2-2-3, or 3-2-2, or 2-3-2). 5/4 or 5/8 is a group of two and a group of three.
But ultimately time signatures are just a convenient way to hint at the beat pattern expected, and to give a framework in which to write the musical notation. The actual _music_ is not so easily transcribed.
It's not always easy to pick out what specific signature is being played at every point in the song, but the "suspended" feeling of hanging beats as the signature changes is very obvious.
For example, a 2/2 tempo with 100 BPM is easier to "manage" than a 4/4 tempo with 200 BPM, and they are the same as each 2/2 beat corresponds to two 4/4 beats. So a "Presto" or "Prestissimo" will often be written as 2/2, with half the BPM that it would have in 4/4. Likewise, a 6/4 is usually faster than a 6/8 or 12/8.
So: the time signature just describes how many times to count within a bar/measure, and what subdivision (that corresponds to a certain notation) you are counting. For communicating to another musician the rhythm of a song you'd just drop the subdivision part, because it's only relevant to notation.
You would lose something by doing so; when a song is written in 6 it's because (unless someone is playing silly buggers) it has some kind of 1-2 pattern. For a lot of songs in 6, if you want to count something smaller you're better off counting it in 2s (with some level of triplet-pervasiveness).
3/4 is ONE two three, with a foot stomp on the ONE. It's a dance beat - STOMP skip skip.
4/4 is so common it appears in a lot of variations, from the bass drum beat of EDM to all the different rock and pop variations. The usual - not infallible, but strongly suggestive - tell is how often the chords change. Failing which, there's often a unique rhythm marker at the starter of each bar.
6/8 is skiddledeediddledee - often used in folk and/or ethnic music. It's a much lighter, more skittery feel.
You can generalise the /8 time sigs to 9/8 and 12/8, although they're more rare. They're often used to signify something exotic, alien, ethnic, or maybe a bit spiritual.
Like this, which you've probably heard, and is in 12/8 when it eventually gets going. The clue is in the phrase length:
https://www.youtube.com/watch?v=jLnL63cXmD8
Exotic sigs like 5/4 and 7/4 are much less standardised, so they're open to interpretation. You can play 5/4 as 2+3 or 3+2 or 4+1. Or just as 5.
Likewise 7/4 (listen to the snare):
How can you talk about feet stomping when that's not part of the music? Two people could chose to stomp their feet on different beats or on more or fewer beats.
There are two common forms of 6/8 (jigs) at least in the kinds of traditional music I'm used to: single jigs and double jigs. A double jig is "skiddledeediddledee" as you said, like the song "99 bottles of beer on the wall", while a single jig is more like "humpty dumpty".
So 6/8 is a variant of 2/4, with 2 main beats in the bar. 3/4 has 3 beats to a bar.
Or in quavers, 6/8 is Da du du Da du du, whilst 3/4 is Da du Da du Da du.
Basically you try to hear what groups of notes are "played as one". You also hear how the accents are distributed among the notes. It's easier to get the hang of it if you have to actually adhere to it when playing an instrument.
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...
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.
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"
I might almost call it 6/4, which wouldn't exactly be wrong (as long as you double all your note values when notating). But that's conventionally reserved for unusual beat groupings like 4/4 + 2/4.
6/4 is extremely common when notating Renaissance music, btw. Although the originals are often in 6/2. Or lack bar lines, so they're X/2.
You might argue that anything in "waltz time" is a waltz, but then your argument becomes circular. You would be saying it's a waltz because it's in 3/4 time and it's in 3/4 time because it's a waltz.
For this piece, I think it's a waltz because it's in 3/4, at a typical waltz tempo, and it "feels like" a waltz, which is to say that the emphasis on the beats and their precise timing feels like waltz music usually does.
So no, I would not argue that anything in "waltz time" is a waltz.
Just as a sanity check, I decided to look up some sheet music and see how others have interpreted the song. Every single one that I can find notates it in 6/8 (interestingly with some individual measures in 9/8).
http://www.musiclassroom.com/partitions/Seal_kiss_from_a_ros...
https://www.musicnotes.com/sheetmusic/mtd.asp?ppn=MN0159794
https://www.musicnotes.com/sheetmusic/mtd.asp?ppn=MN0078268
https://www.sheetmusicplus.com/title/kiss-from-a-rose-satb-a...
https://www.onlinepianist.com/sheet-music/seal/kiss-from-a-r...
https://www.sheetmusicdirect.us/sheetmusic/song/1000176568/k...
If you write it in 6/8 instead, many musicians will emphasize 4 less than 1.
I think that dancers knowing music and musicians knowing dance is extremely useful. It changes your opinions about notation.
Whether it makes more sense to musicians write it 6/8 or not, I have no idea, but it's still a waltz.
Possibly it makes more sense to musicians to notate it as 6/8 because the triples come in pairs. It's still a waltz though, and dancers' instinct will be to break it into triples. It's not hard to scramble a 6/8 signature to make it not a waltz (don't put downbeats every three, and it isn't a waltz) but that song definitely doesn't do that.