c major starting in a different place but using the white notes of the piano is a dorian

No. it is the Aeolian mode aka the relative minor, though the matter of the relative minor is more complex because there are variants of the minor scale related to the harmonic structure of western classical music*. The dorian mode based on the notes of the C major scale (ie the white notes of the piano) is D dorian.

* Those of you who like to correct these things please note. I know there is more to it than I have said here, but I am trying to keep things brief for now.