In honor of the Moon/Earth-orbit analogy, and "I don't wanna get past this for the moment", I'll try and keep this reply focused. :) By the way, if you want to learn foundation-first (Moon/Earth-orbit-style), I'd recommend this: http://voices.azurewebsites.net/elements/notes.htm
. I worked long and hard getting the concepts in optimal order for that.
> As far as I've been able to determine "3rd" is used in at least 3 ways.
Lots of words in natural languages are overloaded, leading to ambiguity. But that's generally because folks are lazy with qualifiers. If I say "an interval of a third" then there's only one thing I can mean. If I say "the third note of a scale" then there's only one thing I can mean.
> 1 - the 3rd NOTE of a scale.
Yes, but that's just using the word as an ordinal number. There's nothing specifically musical about using the word in that context. So, I wouldn't count this is "a way that the word 'third' is used in music". I'd just say that's "a way that ordinal numbers are used by people". Is the word "second" in "second fiddle" meant musically? Or is it just an everyday word used in a musical context? I think it's the latter, and I think this is another example of that. It's true that the interval from the tonic to the "third" degree of a heptatonic scale is a third (and it's a major third if the scale is major), but when you get to pentatonic scales, the "third" degree of the scale is certainly NOT a third above the root. So, I wouldn't read anything more into this use of the word as just a plain ol' ordinal number, any more than I'd rely on the house number of the "third" house on the street being an offset of 2 lots away from the first. It might be; it might not (if there are vacant lots). Personally, I don't find the word to be useful in this context. The "third note of the scale" isn't as useful IMHO, as giving me its interval encoded as a cardinal number. For example, the degrees of the major heptatonic can be encoded as "1 2 3 4 5 6 7". The major pentatonic as "1 2 3 5 6". In both cases, the third degree is a third above the tonic, but the fourth degree, for example, is a fifth above the tonic for the pentatonic.
> 2 - the pitch-distance from the 1st to the 3rd note...
Yes, an interval is an offset. Being computer savvy, you know about bases and offsets. An interval is an offset from a base. Confusingly, the terms used are 1-based, when IMHO it'd be clearer if they were 0-based. So "first" is "no offset at all; stay on the note letter you're on". IMHO, that should be a "zeroth". Intervals use ordinal numbers to indicate the quantity only. That ordinal number is a sequencing of the note letters involved. So, for C to E, you fill in the blanks to get C-D-E, then you assign successive 1-based ordinal numbers to the note letters to get first-second-third, and then the largest ordinal number is the quantity of the interval you have. You still don't know its quality yet, though.
3 - The sound of a 2-note chord, namely of the 1st
I wouldn't say that the sound is itself a "third". The sound of a cow is not "a cow". It's "the sound of a cow". The sound of a third is not a third. It's "the sound of a third". So, again, I wouldn't count this as "a way that the word 'third' is used in music". I'd say it's an example of #2 being used in a sentence. As in, "Hey, please play C natural and E natural together for me. Yep, that sounds like a third to me."
> "THE" major 3rd
To me, this can only mean "the major third as a concept". The major third as a concept is: a musical offset exactly four semitones in size that spans three note letters, inclusive. "A" major third would be some application of that concept. For example, in a major heptatonic scale, there's "a" major third between 1 and 3. Another between 4 and 6, and another between 5 and 7. They're different instances of the same class.
> THE Hz delta between the 1st note of a major ...
It's not the frequency delta, since that'd be an absolute value. It's a ratio. Remember that the frequency curve is logarithmic.
> THE sound of the above simultanously or arpeggiated
That's a sound. Not an interval. I'm getting philosophical here, but only because I'm expressing what my own understanding is. Not trying to influence. If folks want to denote "the sound of a third" as "a third", then that's fine. :) I can't do that.
PS, looking at your diagram, note that major and minor don't apply to fourths and fifths. They're perfect, or they're diminished/augmented 1 or more times.