Scale alterations is definately the most useful way to approach them.
There's no difference between the notes of modes - the difference is the relationship of those notes to the tonal center. So if you're in F Lydian or C major, you're playing the exact same tones... it's what you do with them that makes all the difference.
The explanations like "F Lydian is C major starting from F" are accurate in that they describe the notes - but then you're left playing C major stuff (that's where your ear will lead you, because major tonalities are so familiar) and wondering why you don't sound Lydian.
Now if you start to play in F, you'll be making F the tonal center - for the same natural reasons you'd make C the tonal center if you're 'thinking' of F Lydian as a C-type scale. Now that you're in F, you simply raise the fourth, making all those Bb notes into b naturals, and presto - you're soloing in F Lydian.
Lots of scales have more than one name, because lots of scales sprang up independently in different places. Japan, China, Indonesia - they all have what we'd call the pentatonic minor, but they have their own names for it.
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