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.

I'm working on the spellings to be as accurate as possible :)

I didn't mean to be so harsh here: it's more like the pain that classical singers feel when they hear vowels pronounced "incorrectly" on the radio :-)

Haha i understand :)

By "min 3rd" you mean "minor 3rd", that is, an interval with 3 semitones?

E.g., an example of a minor 3rd would be an A and the first C above that A? E.g., on violin could play the A the first A on the D string and the C the first C on A string?

Yes, my preferred way of looking at chords is superimposed intervals. The four chord qualities (major, minor, augmented, diminished) are built like:

- Diminished: Minor Third + Minor Third

- Minor: Minor 3rd + Major Third

- Major: Major 3rd + Minor Third

- Augmented: Major Third + Major Third

With the intervals being between the tonic and third, third and fifth. So for your example, that would be A, C, Eb as a diminished triad.