One of my favorite examples of this is Dreams by Fleetwood Mac, but that may be because I've been practicing it on harmonica.

The song begins (with note names and relative major scale numbers):

    Now here you go again
    A   C    D   E  D C
    6   1    2   3  2 1
  (low)

But that's not how Stevie Nicks sings it. She starts the word "go" very flat, not quite all the way down to the Eb, but maybe 70 cents down from the E? And then she slides up to the E from there. Listen to her sing it and you will know what I'm talking about.

If you have an instrument where you can play either the straight notes or the notes with that slide, play it both ways and you will hear the difference.

This song is fun and easy to play on harmonica, including the pitch bends, if you get or make the right harp. You need a High C Melody Maker, which you can order from Seydel or you can take a Lee Oskar F Major and retune it into a C Melody Maker: raise the blow 3 from C to D, and raise the draw 5 and draw 9 from Bb to B.

How do you retune a harmonica? You file the reeds! Take off some weight near the free end of a reed and it will vibrate faster, giving it a higher pitch. More strangely, you can also lower the pitch a little by thinning the reed near the fixed end.

I mean if you move down 3 notes from any Major key you have your Minor key with the same exact notes, like C Major and A Minor.

I usually compose with a Major or Minor key in mind, but it's kind of a reference and not a be all end all. I like sharp 4's which work great usually and don't fit in either key. A flat 7 sounds great in a major key too, usually. I agree with you mostly, and i am annoyed by people who are adamant about the rules of a key, but it helps me to have a key in mind to both play and compose, even if i break the rules of it.

I find improvising on piano / guitar is nearly impossible if you don't know the key, even after many years of playing both.

If you think in terms of modes, then using a sharp 4 in major you can think of as lydian (the 4th mode of major) or lydian dominant (has a flat 7 too, the 4th mode of melodic minor). In minor, there's lydian minor (the 4th mode of harmonic major).

Major but with a flat 7 is mixolydian (the fifth mode of major). It is everywhere in popular music.

My favourite thing at the moment is major 7 sharp 5 chords (or an augmented triad with also a major 7th). These chords arise in harmonic and melodic minor (in both cases chord 3), and in harmonic major (chord 6). But they also sound great as substitutions for what is 'supposed' to be a major 7 chord.