Yeah, that is an odd one. It's not commonly recognized as a "standard" designation for a chord. I see the logic in using this as a shorthand for a fully-diminished seventh chord, because that would follow the conventions of other chords. After all, "C" by itself indicates a major triad, and "C6" means the C major triad + the 6th. It would seem to follow, then, that "Cdim" is a triad (which is true) and so "Cdim6" would mean to add the same 6th (A).
However, this is not the common way to describe this chord. I can think of at least one point of confusion: if you say "Cdim6" someone might assume it means "C major with a diminished 6." A diminished 6th is practically never, ever used, but very technically speaking it would be A double-flat in the key of C. In other words, the "diminished 6th" sounds just like the 5th (G), similar to how the diminished seventh (B double flat) sounds like the 6th (A).
In looking back at the original question, is there any chance that the chord on the paper was actually a "Dmi6" chord? This isn't super-common, but I've seen "mi" sometimes used to mean "minor," making "Dmi6" a D minor 6th chord.
Just my 2 cents.
James
