It's funny how we all have different ways to understand music, Allan Holdsworth even invented his own rules and notation... I was educated in this field by Yury Pronin, a russian maestro (who studied with Dimitri Shostakovich) he said to me that in reality the important thing about chords is their "function", there are three kinds of functions: Fundamental, Subdominant and Dominant, a chord can mutate its function when another tension is added (i.e. in C major, the diminished triad b-d-f has a dominant function, but when the seventh is added it mutates to a subdominant function: b-d-f-a). Some times a chord can have two functions, (i.e. Em7, or any third grade in a mayor scale can be fundamental or dominant), Also when modulating to another tonality a chord has two functions, one to the key where it comes and another to the following key. This way of thinking is based on composition, where you are sure that nobody is going to add tensions or added notes to your written chords.
In jazz even if a Dm7 is notated you must take in mind that the pianist or guitarist probably is going to play more tensions of the chord so you must be aware of the whole scale corresponding to that chord all the time.
I like to teach the functions method starting with triads to beginners, they get the picture very clear and simple very fast. Twist and Shout!!