They’re really all the same thing, just different ways of looking at it. I think looking at all the ways give a better understanding of what modes are, where they come from and how you can use them. Parroting out the formula of the modes in relationship to the major scale isn’t really that useful. Knowing that D Dorian is the C major scale starting on the second note helps you derive the modes and shows you where they come from but it’s not that helpful if you need C mixolydian that you need to start on the 5th note of the F major scale except maybe that if you know key signatures, you know that F has 1 flat. I just remember that Mixolydian goes with a V7 chord.
All I try and remember is the pnemonic “I Don’t Particularly Like Modes A Lot” and which modes go with which chord types
I was trying to think back over the songs I have posted here recently to see if any of them used any mode other than Ionian or Aeolian at all. I came up with one, out of probably dozens of songs; it is Mixolydian, and probably because Hooky thought “This note sounds better!” rather than “I’m gonna write this in Mixolydian! Suck it, Reddit!”
Pardon me if this has already been suggested, but…
Learn the shape in each finger position (2nd finger, common to starting the way we learn the C major, or any major scale, the 1st finger position, common to starting the minor scale, and the 4th finger position).
Once you know how to play the shape in any of those patterns, you can find the note you want to start on, and easily play that mode.
Hopefully, by learning those shapes you will also begin to recognize them by intervals, and if you don’t immediately start to recognize the modes shapes by intervals, then, there is some more homework for you that can be very valuable to learn.
Yes, you will technically be learning the modes of the scale, be it the C Major or G or F#, you will be able to transform to playing the right mode simply by playing the pattens from the correct starting note
This is actually the best part. Once you get that all the modes are related, it’s all one repeating interval pattern, and the intervals are all that matter - that’s when it starts to feel really cool.
It even helps a lot when you realize that you only need to memorize two things - the major scale interval pattern, and the order of the scale degrees. Then you can figure out any of the modes trivially by rotation. Once you see the interval pattern and how it rotates it all makes sense.
This is what I was getting at about why I think software engineers might be predisposed to love this stuff. This is almost exactly how binary coded arithmetic works for the rotation operator.
I’ve written that out many times it also makes it fairly obvious that Dorian is a musical palindrome and which others are the inverse of each other.
Dorian is also the neutral point for “brightness”
It’s interesting/useful to observe that the brightness of the modes follows the circle of fifths (and the order of sharps) from F Lydian (brightest) to B locrian (darkest) in C major🙂
why you gotta be that way? 
it also makes it fairly obvious that Dorian is a musical palindrome and which others are the inverse of each other.