In a diminished triad the diminished refers to the fifth of the chord and the third in understood to be a minor third.
In a diminished seventh chord the diminished refers to the seventh of the chord and the third is understood to be a minor third and the fifth is understood to be diminished.
The names are not a complete list of the intervals of the chords.
Think of it like the name major scale , not all the intervals in that scale are major, it's just a name, you still need to learn the interval structure.
To some degree the simple names work because the interval structure is understood relative to the diatonic scale.
Major/minor triad refers to the third only and the fifth is assumed perfect, because all fifths are perfect except the one above the leading tone, and so the triad built on the leading tone is called diminished referring to the unique fifth in only that one diatonic triad.
In the case of non-diatonic seventh chords, the naming becomes more explicit, sort of listing out intervals.
For example a minor major seventh chord is a minor triad with a major seventh added.
Why do diminished chords have 3rd minor?
Because the thirds of the triads built on the leading tone of diatonic scales, or the second degree on minor diatonic scales, are minor thirds.
music.stackexchange.com/users/23919/Michael Curtis
A diminished chord has a diminished fifth , not diminished third https://en.wikipedia.org/wiki/Diminished_triad Diminished third is two semitones
https://en.wikipedia.org/wiki/Diminished_third "root + 3rd diminished + 5th diminished" ...
if you play that rooted on the first scale degree, it gives a Lydian'ish sound, because it's enharmonic to something like II 7 / I.
music.stackexchange.com/users/51766/piiperi Reinstate Monica
Let's say the root is F#, for example.
The diminished chord would be F
# A C. (If it were a diminished 7th chord, it would be F# A C Eb.)
The diminished third would be Ab.
That would make the chord F# Ab C (plus Eb if you add the diminished seventh).
That chord exists: it's the German sixth chord.
music.stackexchange.com/users/29126/Rosie F
A minor third is unambiguous, and generally accepted to be consonant ("pleasant sounding").
By using a diminished third, we are dangerously closing in to a non-third-based chord, which are generally regarded as too dissonant for basic harmony (i.e. a different area or level of expertise).
A diminished third is more similar to a "second" interval.
music.stackexchange.com/users/9563/Hatebit
If we define the triads of all degrees of any scale (minor or major) we have major thirds and minor thirds with perfect fifths: the terms major / minor refer to the lower third.
Only the seventh degree ti-re-fa (= B,D,F) in a major scale and the second degree of a minor scale se-ti-fa (= G#,B,D) has a diminished fifth -
that's why this chords are named "diminished": the term diminished refers to the dim.
5th.
(In a chromatic minor scale the 3rd degree can contain 2 major thirds (do-mi-se = C,E,G#).
This triad is named "augmented" referring to the augmented fifth (+5 or #5).
"Double diminished" would mean that both intervals (3rd and 5th) are diminished:
double diminished refers to the third and the fifth!
(so double means that both intervals are diminished ).
music.stackexchange.com/users/56219/Albrecht Hügli
There isn't always a completely logical reason why things are named as they are.
A diminished triad has minor 3rd and diminished 5th A diminished 7th chord has minor 3rd, diminished 5th and diminished 7th.
I suggest you accept this and move on!
A diminished 3rd is theoretically possible.
(So is an augmented 7th.)
But they sound so like a major 2nd (and an octave) that there's rarely any point in spelling them that way.
And I don't think you'll gain anything from worrying about such extreme cases.
Root, diminished 3rd and diminished 5th (C, E&flat;&flat;, G&flat;) will be heard as C, D, F&sharp;, a bare-bones D7.
music.stackexchange.com/users/17003/Laurence Payne
Why do diminished chords have 3rd minor , not 3rd diminished ?  and what is the name of this chord : root + 3rd diminished + 5th diminished?
Why do diminished chords have 3rd minor?
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music.stackexchange.com/questions/100379
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S5: Major/minor triad refers to the third only and the fifth is assumed perfect, because all fifths are perfect except the one above the leading tone, and so the triad built on the leading tone is called diminished referring to the unique fifth in only that one diatonic triad.   In the case of non-diatonic seventh chords, the naming becomes more explicit, sort of listing out intervals. If we define the triads of all degrees of any scale (minor or major) we have major thirds and minor thirds with perfect fifths: the terms  major  /  minor  refer to the lower third. Only the seventh degree ti-re-fa (= B,D,F) in a major scale and the second degree of a minor scale se-ti-fa (= G#,B,D) has a diminished fifth - that's why  this chords are named "diminished": the term  diminished  refers to the dim. 5th A minor third is unambiguous, and generally accepted to be consonant. 
Major/minor triad refers to the third only. In the case of non-diatonic seventh chords, the naming becomes more explicit. A minor third is unambiguous, and generally accepted to be consonant. By using a diminished third, we are dangerously closing in to a non-third-based chord. If we define the triads of all degrees of any scale (minor or major) we have major thirds and minor thirds with perfect fifths. Only the seventh degree ti-re-fa in a major scale and the second degree of a minor scale have a diminished fifth.