Taking C major/C Ionian as a start point, as in C D E F G A B C, the mode starting on note 2, D, using all the same notes, will be D Dorian. This is sometimes, confusingly, called 'the Dorian of C'. You , I think, are finding 'the Dorian of Bb', which will start on C, and use the notes from Bb maj/ Bb Ionian.Thus, the notes involved will be C D Eb F G A Bb C.
With modes, I feel it's more straightforward to refer back to the 'parent key'. As in, for example, C ionian- C maj. D Dorian - C maj. E Phrygian - C maj. F Lydian - C maj. G mixolydian - C maj. A Aeolian - C maj. B Locrian -C maj.Each of these modes uses exactly the same notes as each other - as in C maj.
Rather than working out the tone/semitone spacing for each mode, it's done for you if you refer back.Looking at the 8 notes in an octave, spread round a circle, starting at a different point will give you a different mode.This obviously works for each and every major, just need to put in the appropriate # or b. E.g. G Dorian comes from F maj., and will have a Bb in it.