Tonality is probably the closest to an overarching term that I know for what you’re asking.
major, minor, diminished, augmented, and the symbols for flat and sharp are used to represent the interval between the root and the 3rd, 5th, 7th, 9th, 11th and 13th of a 2octave scale. Which interval is being specified varies with the chord
One thing to understand is that written chord charts are trying to convey as much as possible with as little as possible so it’s expected that you know certain things that aren’t written down.
for example, traditionally, major triads are the most common chord so
C is all you see to know you have a major 3rd and a perfect 5th C-E-G
minor chords are next so
Cm (lowercase m) tells you it’s a minor 3rd and a perfect 5th C-Eb-G
dominant 7ths come next so
C7 tells you major 3rd, perfect 5th, minor 7th C-E-G-Bb
Cm7 minor 3rd, perfect 5th, minor 7th C-Eb-G-Bb
diminished is a little bit tricky. the interval between the root and the 3rd is described as being either minor or major C to Eb, or E. The interval between the root and the 5th is described as being either diminished, perfect, or augmented as in C to Gb, G, G# respectively, and the interval between the root and the 7th is described as being either minor or major, C to Bb, or B. But a diminished triad refers to a minor 3rd with a dimished 5th: C-Eb-Gb, and a diminished 7th chord is C-Eb-Gb-Bbb(A).
Therefore Cflat5 or Cb5 would be written to express the chord C-E-Gb in which only the 5th interval would be diminished
Dominant is another term for tonality, this chord contains a major 3rd and a minor 7th counting up from the root, thus
but the interval between the 3rd and the 7th, E-Bb is a diminished 5th, which is what gives the chord it’s feeling of drive and tension, in wants to go somewhere, usually home to the home chord.
Jazz musicians often think of chords as having only 3 tonalities, minor, major and dominant and feeling free to substitute chord within each group, thus
C-major family-C6, C6/9, Cmaj7, Cmaj9, all these chords have a major 3rd
Cm-minor family-Cm7, Cm9, Cm11, all these chords have a minor 3rd
C7-dominant family- C9,C11,C7b9, C9b5, C7#5b9(a personal favorite) all these have a major 3rd and a minor 7th.
Clear as mud? patience!