It seems like you are aware of the fact that the modes are related to the major scale and also aware of the correlation between these modes and the chords that occur in a given major key. This is an interesting question. It is not false to say that if you are playing C major you are also playing D dorian and/or A minor. But how one actually hears the diatonic patterns and sequences will determine what mode the claim to be listening to. When we make music we establish patterns by playing "key tones", tonal centers of chords, on strong beats (at least that is a formula in many Jazz improv pamphlets). So if you were to play a simple melodic phrase (C, G, F, E, D, E) in the key of C, or (1, 5, 4, 3, 2, 3) and keep repeating it over each new chord in the progression it will always be a C major melody and perceived as such. You will hear interesting contrasts between this and the notes of the each new chord as the melody will (1) align with the tonal center of some chords but not others (creating more/less dissonance), (2) have chord tones on weak beats rather that strong beats even when they do align, (3) create interesting extensions of the chords, and (4) generate a variety of harmonies with the chord tones. If, on the other hand you repeat this phrase over C, Dmin, and Emin, then move it up to the the 4th on F (F, C, B, A, G, B) (notice the #4 relative to F) then I'd say you have been in C maj for three chords and are now in F lydian. Changing the chord under the melody does not necessarily mean you have changed mode. But if you move a motif in a way that places tonal centers of the chord on strong beats (or uses some other device to indicate a change) then it would be appropriate to say you've changed modes.
On another note one can play through all 7 modes on one chord. I do this over So What by Miles Davis. Even though the song is a vamp on D-7 (in the key of C) and has a completely D Dorian feel (as indicated by the Bass line and all the solos), I like to play with melodies that are explicitly on C maj. The C Maj7 arpeggio has the (b7, 9, 11, 13) of the D-7 and these are great extensions. So, am I in D Dorian because I am playing over D-7? Or am I in C maj because I am explicitly playing with a C Maj7 motif? There may be no clean and decisive answer to that. I would say it depends on the motif I choose, the order I choose to play the C Maj7 arpeggio and whether I land on the C or E (b7 or 9) or the other tones as those will be more likely to conflict (especially the 4th of D-7). I think of it as being in C major when I do this because I am trying to "force" C maj tonality over the D-7 chord. It works but you do hear the "force", the tension of fitting two chords together.
One thing is for sure, changing chord does not equate to changing mode.