When I hear greensleeves it sounds like a waltz to me (bom-tah-tah / bom-tah-tah), why is it considered a sextuple ?
In simple terms, 6/8 is compound !! You are counting a relatively quick 1-2-3-1-2-3, as you say, which can feel like 3/4 or 3/8 - waltz like. However, the second '1' is slightly less emphasised in 6/8, so it's counted ONE-two-three four-five-six. (The four is where your second 'one', or 'bom' is).
6/8, being compound, means that it can simultaneously be counted in 2 time, as ONE-tah-tah TWO-tah-tah. Each syllable coinciding with one of the faster counts in the earlier example. If the second 'bom' is as loud as the first 'bom' when you count, it will be in 3/4 or 3/8 time, but when the second 'bom' is quieter, but still louder than the other beats, the song will be in 6/8.
Music theory generally uses only duple and triple, not the quadruple and sextuple that your link mentions. Any meter can be explained in terms of duple or triple along with simple or compound. For example, the "sextuple" meter she mentions is compound duple meter.
It works like this. Meter is a combination of strong and weak beats. Two or a multiple of two strong beats per measure is duple, three or a multiple of three strong beats per measure is triple. If there is only one strong beat per measure (2/4, 3/4), then all the beats count. As for simple vs. compound, strong beats that subdivide into two are simple, those that subdivide into three are compound. 2/4 and 3/4 subdivided in twos by default.
So, for the commonest time signatures:
2/2 Simple Duple
2/4 Simple Duple
3/4 Simple Triple
4/4 Simple Duple
6/8 Compound Duple
9/8 Compound Triple
12/8 Compound Duple
The fact that some strong beats are weaker than others makes no difference in whether a meter is compound or simple or duple or triple.
If you are having trouble with, say, the difference between 2/4 and 4/4, think about primary and secondary accents on syllables. For example, "secondary" has a secondary accent on the third syllable: "SE-con-Da-ry". In musical meter, the primary accent is on the first beat, and the secondary accent is on the other strong beat (if there is one). In 4/4, the secondary accent is on 3.
Because it is in 6/8 and while you feel it in three, you feel it in two groups of three as is common for a song in 6/8. In 6/8 two dotted quarter notes get the beat, but each dotted quarter note can further be divided into 3 eighth notes making you feel each beat in 3. That is how songs in 6/8 typically works.
The distinction between a slow 6/8 meter and a waltz is largely arbitrary. It may be that the melody is intended to have a stronger downbeat on every alternate bar, and that many people perceive it that way, but the same is true for countless waltzes notated in 3/4 as well (mostly those that have even-length numbers of bars in their phrases). The only objective reason to call something 3/4 if it has an odd number of bars. (For a fast 6/8 meter, such as a typical tarantella, it's the short duration of the quavers that makes it much more appropriate to use 6/8, but "Greensleeves" isn't one of those at all.)
Every copy I've seen of Greensleeves notates it in 3/4 time, but if one looks at the word stresses it could probably be notated just as well in 6/8 (with half as many measures) or perhaps 12/8 (with a quarter as many measures) since alternating measures definitely have stronger and weaker word stresses, as do alternating pairs of measures (though to a lesser degree). I think that historically there has been a tendency to avoid the use of compound meters for slower tempos, even though such avoidance often results in notes being held routinely for entire measures. Greensleeves doesn't have any really long notes, but consider the song "On top of Old Smokey". All the sheet music I've seen for that is likewise in 3/4, but each four-measure phrase has two notes which is held for a whole measure each, and the downbeat of the third measure is a rest.
The article you linked cites a nice example of a piece which is in triple meter without any sort of sextuple feel: Brahms' Lullaby. In the English text "Lul-la-BY, and good NIGHT; in the SKY stars are BRIGHT", the stresses on "-by", "night", "sky", and "bright" are all roughly equal (interestingly, the "triple" example where it shows word stresses, "America", would fit naturally into a 6/8 meter while Brahm's Lullaby would not).