: Are all diatonic chords in the diminished scale diminished? I think there's some confusion about the definition of the word 'diatonic'. I'm not using it here to refer to the the prominent primal pattern of the heptatonic

The other answers provide a lot of information and go into a lot of technical detail. I could list all the nuts and bolts as well, but it seems like you are just discovering music and want a simpler algorithm to memorise, perhaps.
There are 2 diminished scales, each has 8 notes per octave. You can think of those notes as being odd or even in number (for the C half-whole diminished scale, note 1 would be C, the root, and note 2 would be C#).
In terms of chords, both half-whole and whole-half diminished scales follow a very predictable pattern:

The half-whole diminished scale
Every odd note produces a major triad[1], a minor triad[2] and a diminished chord[3]. Every even note produces just the diminished triad.

The whole-half diminished scale
The pattern gets reversed: now every odd note produces just the dimished triad, and every even note produces a major triad, a minor triad and a diminished chord.

It's that simple!

1 The formula for the major triad is 0-4-7 (where 0 is the root, and every other number denotes how many semitones away from the root the next note is) or R-W+W-W+H (where W is a whole tone and H is a semitone).
2 The formula for the minor triad is 0-3-7 or R-W+H-W+W.
3 The formula for the diminished triad is 0-3-6 or R-W+H-W+H.

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If you consider every chord made up of notes from a diminished scale to be "of the scale", then a diminished scale contains not just diminished triads. The diminished scale is highly symmetrical, so to find triads in a diminished scale you only have to consider a small number of candidates, which you can then transpose to get the full list.

Let's look at the C half-whole diminished scale:

C C# D Eb E F F# G Ab A Bb B
* * * * * * * *

We'll only look at triads starting on C or C#:

C C# D Eb E F F# G Ab A Bb B
* * * (*) Cdim
* * * Cm
* * * C
* * * (*) C#dim

To get the full list of triads, we transpose these four chords in steps of three half-tones:

Cdim Ebdim F#dim Adim
Cm Ebm F#m Am
C Eb F# A
C#dim Edim Gdim Bbdim

This is the list of 16 triads for C half-whole diminished, but also for:

C half-whole diminished
Eb half-whole diminished
F# half-whole diminished
A half-whole diminished
C# whole-half diminished
E whole-half diminished
G whole-half diminished
Bb whole-half diminished

because these eight diminished scales are enharmonic, they are made up of the same notes. There are two further "families" of diminished scales: the scales above transposed 1 half-tone step up, and the scales above transposed 1 half-tone step down.

To find the 16 triads for a diminished scale from one of the other two "families", check whether it is a scale in the list transposed a half-tone up or down, and then transpose the 16 triads a half-tone up or down.

You will notice that the list of 16 triads for one "family" of enharmonic diminished scales contains 4 major triads, 4 minor triads, and 8 diminished triads. This means that two non-enharmonic diminished scales share 4 diminished triads, but no major or minor triads. E.g. the C half-whole diminished scale and the C# half-whole diminished scale share these triads:

C#dim Edim Gdim Bbdim

while the C half-whole diminished scale and the B half-whole diminished scale share these triads:

Cdim Ebdim F#dim Adim

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This question seems to hinge on how things are spelled - including enharmonic spellings - and the effect of spelling on interval names.

If the diminished scales are spelled as two superimposed diminished seventh chords, we get...

|C |Eb |Gb |Bbb |
| D| F| Ab| Cb|

...or...

|C |Eb |Gb |Bbb |
| Db| Fb| Abb| Cbb|

Let's use the second scale for examples. We can make a chord C Fb Abb and enharmonically re-spell it to C E G. The first chord C Fb Abb is a diminished fourth and a minor third. The other spelling is a major third and a minor third.

We could re-spell the scale so that the two superimposed diminished seventh chords structure is obscured:

|C |Eb |F# |A |
| Db| E| G| Bb|

With that spelling we can make C E G without any enharmonic re-spellings.

Of course the issue here is the gamut (the seven letters) are for a heptatonic diatonic scale, but the diminished scales are octatonic, and the naming of intervals is first based on the position in the letter sequence and then qualified by the size in semi-tones.

When tertian chords are built the spelling details matter. C E G is two stacked thirds where the specific qualities are major third and minor third. C Fb Abb is NOT two stacked thirds. It is a diminished fourth and a minor third. If our "diatonic" chords are to be tertian triads in root position, then C Fb Abb is not a diatonic chord, but C Eb Gb or Db Fb Abb would be diatonic chords.

For what it's worth, when the second scale of superimposed diminished seventh chords is used and we have chord like Abb Cbb Db it's is enharmonically equivalent to a diminished triad, but as spelled it is an inverted diminished seventh chord with the third tone omitted.

Are all diatonic [of the scale] chords in the diminished scale diminished?

I think "yes" provided that the chords are tertian and the scale is spelled as two superimposed seventh chords.

EDIT

Part of the discussion with this question is about the possibility to construct a major triad in the diminished scale and should that be considered diatonic.

In my initial answer where the diminished scales were constructed from superimposed diminished seventh chords I said "no" because the chord spelling used a diminished fourth.

However, you could construct the diminished scale using superimposed major triads and get different spellings...

scale degree: 1 2 3 4 5 6 7 8 1 2 3 4
C Db Eb E Gb G A Bb C Db Eb E
| (C#)| | | | | | (C#) |
---------------- | | | |
| | | | | | |
---------------- | |
| | | | |
---------------- |
| | |
----------------

The three chords are C, Eb, Gb, and A all major triads in a sequence of chromatic mediants. The A chord needs an enharmonic respelling of Db to C#.

You could then harmonize the scale with a series of alternating root and first inversion triads:

C A6/3 Eb C6/3 Gb Eb6/3 A Gb6/3 | C

Now you could say all the 'diatonic' chords of the diminished scale are major triads.

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The diminished scale is often used with the 7b9 family of chords. C half-whole diminished is associated with the C7b9 chord:

C E G Bb Db

(the whole scale is C Db D# E F# G A Bb... notice that you have a major 3rd between C and E thanks to skipping over both Db and D#)

Edim7 is the same chord as C7b9 without C in the bass, so they're often completely interchangeable.

You can create very nice extended chords by adding-in other notes from the C diminished scale:

C E G Bb Db A (C13b9)
C E G Bb Dd F# (C7b9#11)
C E G Bb Db D# (C7b9#9)

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Yes.

The diminished scale is symmetrical: every third is the same distance as all the other thirds. Since all the thirds are minor thirds, all the chords that naturally occur in the scale will be m3+m3 = a diminished triad.

The same will be true of the whole tone scale: every third is a major third, so every triad will be augmented.

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