Yeah, if you play (D, E, F, G, A, B, C) over a Dm chord; then you are indeed playing a D Dorian.
And if you play (D, E, F, G, A, Bb, C) over a Dm Chord; you are playing a D minor Scale or D Aeolian.

Only problem is, you are still thinking about the Major Scale.
And you may get confused at some point if you keep doing that.

So, your reasoning should be something along this lines:
- Ok, I’ve got a Dm Chord playing behind.
- There are a couple of scales I can use over that chord such as: D Dorian,
D Minor Pentatonic, D Blues, D Phrygian, D Harmonic Minor, etc.
- Do I want a bluesy feel or do I want an exotic sound.

The good thing is that you understand now, just try to break away from the habit of thinking about the major scale.
And you will enjoy the unique sound of each mode without having to think what major scale you should play and what degree you should start from.

I know that the easiest way to learn the modes on the guitar is to take a “Major Scale Pattern” and then just start on a different note to play an specific Mode.
But I’ve seen many frustrated guitarists who know the “Mode Patterns” by heart, but don’t know how to use them.

And yes, changing one note makes a huge difference.
Isn’t just one note the difference between a major and a minor chord? :)