IMHO, everything you wrote is pretty much acceptable - that is D Dorian (which is the Dorian mode of the C scale). You're saying "or" but then in my mind saying the same thing. Regarding that D Dorian, as you may know already, the key signature on staff contains no flats or sharps because it's notes from C Major only. C Dorian would contain only notes from B Major (aka Ionian), starting on C.so lets just use the key of c for example
C d e f g a b c is C ionian
D e f g a b c d is D dorian correct?
Is that Mode called D dorian? Or is it the dorian mode of the c scale? Or are you just sposed to know by D dorian that ur in the key of c bc d is the second step of the c major scale?
Second question are the mode "box shapes" the same for both major and minor keys?