Chord naming and interval naming are two pretty different things -- for example, your Db-F#-A-C chord's name would more likely focus on the F#, since you create a minor triad between F#-A-Db(C#). That's all highly contextual, and talking about prime or octave intervals in chord context is extremely rare.
For the interval naming question, my understanding is as follows:
- 2nds, 3rds, 6ths, 7ths have major and minor variations with aug/dim at either extreme.
- 4ths, 5ths, primes, and octaves have a "perfect" variation, with aug/dim, again, to be used outside of this.
So looking at two notes -- D and Db. Depending on the register of each note, this interval is either going to be a prime or an octave of some flavor--both notes are some kind of D.
If the two notes are a half step away from one another, this interval is always called an augmented prime. The interval is one half step larger than a perfect prime, which is a distance of 0 half steps.
In the case that the two notes are in different octaves, suddenly the ordering dictates whether the interval is larger (augmented) or smaller (diminished) than a perfect octave:
- Db in lower octave, D in upper octave is an augmented octave, enharmonic to a minor 9th
- D in lower octave, Db in upper octave is a diminished octave, enharmonic to a major 7th
So to rehash the whole answer here: a "first" in chord talk necessarily is unmodified. If your chord is named X, even if that note in particular is only implied, any "variations" of that pitch would have a name of an upper chord member (9th or 7th). If the variated note truly was functioning as a chord root, then your chord is really bitonal and it never had anything to do with X in the first place.
In interval talk, you can't have a diminished prime because intervals can't have negative distance. You can instead have a diminished octave.