Its both really.
Ditzy and Derpy are the same character, but used for different purposes.
Ditzy Doo is the character that we tend to use for more serious fanfics, art, and music, mostly as Dinky Doo's mother.
Derpy Hooves on the other hand, is our funny, quirky, haphazardous pony who we love to use in the less serious and more comical side of the fandom.
As for officialness, Amy Keating Rogers has used both names and a combination "Derpy Doo" for scripts, Hasbro Marketing Likes Derpy for good reasons.
In the end, it shouldn't matter, personally I like Ditzy "Derpy Hooves" Doo, this way, everyone is happy