(1) The modes are not just „the same notes played in a different order“- they are the same finger
shapes transposed to different positions- big difference is the new relationship between the
intervals to each other and to the Modal Chord Sequence and Finalis being used.
(2) The modes have no tonal center or tonality, they only tend to be maj. or minor and the particular
modal tone (i.e. C note or CMaj. chord) is called a finalis or home / reference tone or chord as
opposed to being called a tonic tone or chord. No matter what mode you are talking about, they
are all ambiguous in this regard. It’s not the brain that does the „focusing“ or that determines a
modes characteristics, it is a very exact and direct musical device. The backing chords called a
„Modal Sequence Motif“ also play a significant role in determining what Mode is being implied
and the appropriate mode that would be suitable.
(3) C-Dorian in simple terms would 1st of all be most closely related to C (Natural) minor (parent
scale) and not C Major, as dorian is a mode that tends towards minor. So in this case with a minor scale structure there would most definitely be accidentals (3 flat notes= 3 / 6 / 7). All you would
have to do to make this dorian is to take the 6th degree which normally would be flat or minor in a
normal (natural) minor scale and raise it 1 half step up, back to it’s major position (leaving you with 2
flats) and you have C-Dorian in it’s primary structure.
There are 3 ways to look at the C Maj. Modes: (A) as Modal Scales with a "parent" maj. or minor scale consisting of 1 altered interval- with the exceptions of Ionian, Aeolian (same interval structure as parent scale, but only up to the 1st octave) and Locrian which is neither major or minor but tends toward the 1/2 diminished scale (due to all the 1/2 tones) and has all of it’s intervals altered, except the 1 and the 4. (B) As the transposition of the C Maj. scale, the finger shapes stay the same BUT the relationship of the intervals to the backing „Modal Seq. Motif“ changes (C) This I would not recommend: As a scale structure that uses notes borrowed from both related and unrelated keys (not to be confused with the relative key, as there are a total 4 different related key signatures: relative, parallel, subdominant and dominant ) + all corresponding maj. keys / scales based on the number of flats or sharps (circle of 5th’s) used by that particular mode (i.e. C-Dorian contains 2 flat notes and could be considered to be a Bb Maj.) with C as the finalis instead of Bb.
As far as „reconciling“ relative and parallel scales with modes:
C-Ionian is also the relative mode to A-Aeolian and C-Aeolian mode can be considered to be the parallel mode to C-Ionian but this is as far as it goes with the the first 7 C Maj. Modes. The function of „relative“ and „parallel“ keys do not play a relevant role in modal function as they do in Maj./minor functional harmony.
„How do I reconcile the modes of a key being described as not having the same notes sometimes and then having the same notes“?
As mentioned before, modes do not have keys. Your "reconciliation" problem may be due to the confusion that all the 1st 7
CMaj. modes (for Guit.) are based on the standard 5 C Maj. scale patterns. The modes based on these 5 positions do share all the same notes that are arranged and starting on the corresponding modal tone BUT this is where the simplicity ends. In order to play any particular mode using all five patterns you then must transpose them and you can then use the finger shapes that either preceeds or follows the transposed finger pattern to be able to play that mode with the intervals being automatically adjusted. This means you have to know your C Maj. scale inside, outside, forwards and backwards to use these modes easily and quickly. Some modes will use their respective related dominant and subdominant keys for the corresponding major key signature and they will in this way be related to that particular mode and share most of the same notes. The other corresponding key type will only be related in terms of how many sharps or flats it contains in relationship to that particular mode.