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|
|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.
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
A all major triads in a sequence of chromatic mediants. The
A chord needs an enharmonic respelling of
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.