broken chords as i commonly use the term refers the technical exercise, sort of like an arpegio however you hit each inversion on your way up to the next octave,
i.e. c-e-g,e-g-c,g-c-e, repeat, etc. etc.
if you're refering modified chords it depends on which tone is changed, i.e you can augment (raise a half step) or diminish (decrease half step), in jazz and pop you will normally see these written as the root chord then either #__(insert tone, i.e. #2 = sharp second), or b___(i.e. b9=flat 9th).
in modern art music it is almost entirely context dependent especially in chords that do not fit traditional tonality you will come across named chors like german sixth chord, neapolitan chords, even work specific such as the 'tristan chord' , scriabin had the Mystic chord or Prometheus chord ( a complex, six-note quartal chord, scale, or pitch collection which served loosely as the harmonic and melodic basis =C, F♯, B♭, E, A, D)