ok I'm trying to get a global understanding of chord/scale relationships here
I've tentatively broken it down to these 3 concepts:

1 - formula of intervals determines the type of chord.
example:
Root + major 3rd + 5th = a major chord
Root + minor 3rd + 5th = minor chord
Root + minor 3rd + flatted 5th = diminished triad
Root + Major 3rd + 5th + Major 7th = Major 7th Chord
Root + 5th = "5 chord"
Etc

2 - root determines the key.

3 - If the root isn't the lowest note in the chord then it's an inversion.
Inversion is really just a matter of timbre preference... because the chord formula remains the same regardless?

is there another principle I'm missing that isn't subordinate to these three?
Are these three equal; or is one or more of these concepts actually subordinate to one of the other 3?
(I find that discerning hierarchy of concepts helps me to grasp material better)

thanks

Also you can add the major key chord formula which determines the specific chords in every key.

Majaor is

I ii ii IV V vi viidim

So in C Major we have C, Dm, Em, F, G(7), Am and B diminished.

Hope this helps

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i don't really have any idea what you just said; but thanks.

Those are simply all the chords in the C Major Key.

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You build chords on every note of the scale (by stacking triads) and thereby get the chords that belong to that key.
Once you know the formula, you can build chords in all keys and write chord progressions in them.

Hope that helps!

To make it easier for you let's look at C major scale.

Notes in C major scale are:
C D E F G A B C

If you will treat each note like a root of possible chord and use only notes form scale it will looks like:
C-E-G (Cmaj - I)
D-F-A (Dmin - ii)
E-G-B (Emin - iii)
F-A-C (Fmaj - IV)
G-B-D (Gmaj - V)
A-C-E (Amin - vi)
B-D-F (Bdim - vii dim)
(simply take every second note)

Now check the relation between notes in each chord. They will be exactly like those described by you in your first post

The easy's way to understand all this is to look at the root note as home base. The 1st ,4th and 5th are major chords every other chord in that scale will be a minor chord. This will apply to every key of the major scale. In the minor scales it will be 1st, 4th , 5th as a minor chord, all the other chords will be major. I hope this helps someone. It's the way I taught myself to remember it.