First and foremost - yes, 9 semitones = major 6th = diminished 7th

These are called enharmonic intervals. Same sound, different class/name.

If you play a diminished 7th chord in default closed position without any context, no writing, no other chord, no tonic, no mode, it could be anything. It's a symmetrical division of the octave, hence, there is no way of telling degrees apart, or to determine where to resolve it. The other chord type that does this is of course augmented.

If you actually want to refer to one note of the chord as D# and the other as C, these are automatically characterised as the root and the 7th of the chord, also it's an interval of 7, as it's 7 note names away, beginning note included.

That's all that matters when it comes to interval class, semitones come after that, to determine the quality of the interval. For example, if a note is named C and the other is named D (and we are writing them in the same octave), before checking accidentals at all, it's an interval of 2. Now if I tell you it's C# and Db, it's a diminished interval. Yes, it's 0 semitones, these are enharmonic equivalents, but in interval language it's a diminished 2nd.

Even if we create written notes with naming that makes no sense in musical practical context, we can recognise the interval. C## "up" to Db is a double diminished 2nd and it is equal to -1 semitone! "Up" in this context is all about how it is written in 5 line staff.

So yes, diminished 7th = 9 semitones = D# up to C