So if I was playing a song in the key of G, and then changed chords to an E (which is the 6th note in the scale of G), does that mean that I should use the Aeolian mode starting with E for creating licks over that E chord?

If so, I think i understand modes... crazy stuff
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Your still playing in G major, if you use any note in the g major scale ( G A B C D E F# ) you'll be fine. If you use the Aeolian mode with E as the root your still playing G major (or G Ionian if you like to call it that). In my opinion (correct me if im wrong) thinking of modes is useless unless your playing over a modal chord progression. You'll just get yourself confused.
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Yeah modes are not relevant here. You can play the E minor scale and you can start on any note you want.
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Agreed, not relevant. The mistake I made and that many guitarists learning modes also make is thinking that an understanding of the relationship of modes to parent scales and the shapes of the modes on the fretboard constitutes an understanding of modes. To most musicians, the use of modes applies strictly to special chord progressions (modal progressions) as mentioned above. These tend to not have an implied tonal center and have to be carefully constructed or they won't sound modal anymore. Don't get me wrong, those shapes are useful and using them to move scales up and down the neck and/or incorporate outside notes into your playing is great. However, truly modal music rerquires those modal progressions which are 99% non-modern music (think Gregorian chant or something like that). Happy playing!
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modes have nothing to do with making licks just because your in Gmaj/Em doesnt mean when you switch to an e CHORD means you need to switch to E aeolian. You can use anyone of those 7 positions to make licks/phrases. youll be in key and ittl sound good.

Tip: what you can do is chord wrap. Dissect every chord you play within the rhythm you solo over and time it correctly and play the notes within that specific chord. ittl spark some creativity.

just because you switch to lydian position (4th) doesnt mean your in the mode of lydian. your just playing the 4 position major scale

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