Let's say I'm playing Dorian D mode. Is my tonic:
C, because Dorian mode is derived from the diatonic scale of C?
Or D, because my tonal center is in D?
The tonic of Dorian D is D.
In fact this is more or less what makes the difference between the modes. Altough they can have exactly the same notes, the function of those notes, how they are used, is different.
In dorian D, D is used as the tonic: i.e. the piece ends with D, the main phrases end in D, the D minor chord is the most important chord...
A tonic is a tonal concept, so it doesn't transfer 100% when you talk about modes as they are not tonal ideas.
The simple answer is yes to be in D Dorian you need to make it sound like you have D as your "tonic" or else you are not in D Dorian. If your tonic is C then you are in C major, not D Dorian.
The tonic is D (d-minor as a chord). The cadence in d-dorian would be
X: 1 M: C K: Cmaj L: 1/2 [Gdgb] [A^cea] | [Ddfa]2
Though I daresay you'll find rather more often the relative dominant C-major used, rather than the actual dominant A-major.
X: 1 M: C K: Cmaj L: 1/2 [Gdgb] [Cegc'] | [Ddfa]2
A good question! I'd say technically in D Dorian, D is the tonic, as that's the 'key' it's in, and the start point. However, musically, it could feel like it wants to 'come home', and depending on how it's sounding, it could come to rest, sounding good, back at C. So then, one could say the tonic is C.