Different dictionaries seem to use different criteria. But I'd imagine the New Grove Dictionary is more authoratative than the Concise Oxford Dictionary on this issue. (I see there's a new edition of the Grove due out this week - maybe someone could post what that says on tye subject.)
Anyway, the point is, whether you call two notes played together a diad or a diadic chord, you shouldn't feel that the sound is incomplete unless you add that third note which defines it as major or minor or whatever. In fact in doing so you are likely to be distorting the music, when it comes from traditions that don't really work that way. Even more to the point, you can be failing to make use of some beautiful sounds.