Answer: There are two uses of the term, "appogiatura" - there's the appogiatura as an ornament, which is any accented dissonance that resolves stepwise into the primary melody and which is often written as a small "grace note," and there's the term as used in counterpoint, where an appoggiatura is an accented tone that doesn't belong in the current sonority, was approached by upward leap, and resolves downward to a tone that does agree.

In either sense of the word, if you do more than one appoggiatura at the same time they can form a chord of their own. If the chord they form is a diminished 7th, that's your appoggiatura diminished seventh.

And you're right: if the diminished seventh is completed by a tone that is in common between the dim7 and the chord of resolution, then this would be like a common-tone diminished 7. And yes, if we're talking about two appoggiaturas combined with that common tone then it's also a double appoggiatura.

Just as not all rectangles are squares, not every double appoggiatura is an appoggiatura diminished 7th chord, that's all.

These terms are useful in that they help us to make sense of all the various musical devices out there - with a knowledge of the term the device gets stored in the right mental location for later retrieval. It's good to know what the terms are supposed to mean, but if you have a passage that seems to work well its classification is less important than its effectiveness.