Good question. There's a little bit about the subject here – http://www.dolmetsch.com/musictheory31.htm#dimsevenths – but no rule about what the chord should be named in any of the situations described there.

I'm also seeing, on other music-theory sites, the mention of half-diminished chords and diminished-fifth chords, without a satisfactory definition of either. Can anyone shed any light on the subject of these two terms?