The triadic diminished chords are labeled dim, but the diminished 7th version does indeed seem to be labeled as a half diminished chord. Maybe a circle isn't being rendered for the 7. (It's a bit painful seeing it spelled as C Eb Gb A rather than C Eb Gb Bbb. If you play a string instrument, you might tune an A differently from a Bbb, so it's not just a theoretical difference.)

Also, if you're seeing sevenths thrown in, it might be because in jazz chords are more for suggesting a scale than the actual pitches of the harmony. At least, that's how I understand it as a classical musician.