Is there more than one (name for the) note that is a diminished 3rd
Nope only one correct answer. D to F is a minor third. Db to Fb is also a minor third. An Diminished interval is one semitone below the minor interval (In this case) while still being a third away from the root note ie F double flat.
If so, wouldn't this ambiguity exist for all of the other intervals?
No ambiguity only one correct answer.
Notes may be played in the same place and have different names. D to E is a second. It cannot be a third even if the notes are played at the same key as some third interval. To determine what a second is you start on the bottom (Bottom being one) and count until you get to the number two. D(1) - E(2).
If you start on a D both a Augmented Unison and a Minor Second will both be played at the same place on the piano but that does not make an Augmented Unison or a Minor Second the same interval.
This confusion often arises from teachers who try and cut corners when teaching intervals by just making it an exercise in counting semitones. This is absolutely the wrong way to teach it as D# and Eb are both three semitones from the note C but are in now way the same interval.