I could write a book on
Chord Naming. Oh, actually, wait, there are about 1000 books on chord naming. There IS some standard, unshakeable system that everyone uses.
Classical music is very strict, and there's absolutely
NO TWO WAYS ABOUT IT.
This is going to be somewhat of a rant, as there is no way that I can explain the entire chord naming system here. It is a complete College course in itself.
When you say "D half-diminished", you are also implicitly saying several things about the chord. It is implied that it is a seventh chord, and in Root position. This is if you're talking about a very specific chord , i.e. the notes D, F, Ab, C, and not just the chord's "quality". If you're talking about the quality,
- These 4 notes could, actually, when in the middle of a piece, be a Bb7 (with an added 9) chord, without the root.
- It could mean an F minor 6th chord with a D root. The chords sound the same when played out of context, but have a totally different meaning in context.
The possibilities are endless, and it would do no good to go too deeply into it here. A good book on chord naming would be more appropriate.
Jazz music, on the other hand, is radically different. Chords are there more as guidelines. When looking for a dominant quality, for example, one might play a diminished chord, which can serve as a dominant in certain situations. So chord naming in jazz is in general more lax than in classical music. If you talk about a half-diminished chord or a minor 7 b5 chord to any jazz musician, though, he should know both, and that they are the same.
In general, half diminished would refer more to the quality of the chord, and minor 7 b5 is more of a description of the notes outlining the chord.
It is very difficult to answer the "accusations" of klash in the short space of the node, what with there being so many different angles and styles to take into consideration. Let me just, as an ending, say that dm7b5 and fm/d are two RADICALLY different chords (even though they are composed of the same notes). The former is a HALF DIMINISHED chord, and the latter is a MINOR CHORD WITH AN ALTERED BASS. That difference is not a difference between classical/jazz music and pop music, but a fundamental difference in chord quality and function.
And C E G Bb is a C dominant seventh chord. No two ways about it. In context, though, it could be an E minor 6 #11, a Gm13(no 5) chord, etc.
As to the difference between "major-minor 7th" and dominant - that's the difference between the "dry name of the chord" i.e. which notes comprise it, and the functionality of it.