Either of those is correct because you are not tying it to a specific key, they are
enharmonic tones.
Getting all music theory 101 on you, if you are playing in playing in say, Eb major (3 flats on the key signature, see below), the
pitch inventory is Eb F G Ab Bb C D, so, while you the pitches are the same as D# major - D# E# F## G# A# B# C## (yes, that is a F double sharp and a C double sharp, which raises each pitch 1 full step), they are 2 different keys. D# is also considered a "theoretical" key because it is rarely used because it is basically a PITA (Eb is much easier to use).
As to why
accidentals (sharps and flats) exist, I guess is just seemed like the easier thing to do instead of giving all 12 tones individual names(? although there is also some physics involved in the way they are separated?), I really don't know. But I think it is easier to keep up with 7 notes with accidentals instead of 12 individual notes.
