Hmm... I was always taught at school that a chord, to be a true chord, had to consist of at least three notes - hence the word triad. If there were only two notes, then the chord was unresolved. The notes A and D can form part of many chords, not just a D major or minor.

Many chords can have several notes in common, of course. Am shares common notes with C6 (for example), and whether you're playing one or the other can depend - as you rightly say - on context, or on voicing.