You can take a look to this article:
It seems that the "d" notation was introduced in order to remove the ambiguity of a mixed instructions (verbal and numerical).
The "d" notation introduces a dice-specific algebra that allows you to find easily the equivalence of dice roll.
roll 30 ...
While die may be the term for a single dice when you say "6-die" it sounds like you are talking about more than a single one. calling a die a "D6" or "D20" is more convenient overall and also easier to say how many of each type you want. "6 D6 and 2 D20" as an example. If you where to say it as "6 6-die and 2 20-...