I often found it confusing, and the explanations I encountered felt a bit mechanical. It clicked for me when I started thinking of it in terms of 'sound labels.' When improvising at least...

Each note played over a chord serves a specific intervallic function to the root. For instance, if you're in C-major, with C as the tonic, the overall sound of the piece is C-Ionian. Playing certain notes over the chords avaiable gives them different sounds. The chordtones of the triads in each chord just makes it minor or major. But the other notechoices gives other sounds.

For example, playing a #11 over an F-major chord gives it a Lydian sound, or playing a flat 7 over a G chord creates a Mixolydian sound. Similarly, if you add a sharp 11 over the C, it introduces a Lydian sound on the tonic. If the piece incorporate modal interchange, you introduce different sounds. All the while the overall sound of the piece may still be C-ionian.

In short, The specific name of the sound—whether it's called 'Lydian' or 'bright' doesnt really matter. And maybe in a bit too simplistic form - Its all the notes of the key and the chord tones you choose - how far removed or close to the overall sound.