Yes, you've identified the pronunciation of the vowels correctly.
I suppose singers simply pronounce things the way they like the sound of. I don't think there's any special rules of phonology.
The only thing I might say, specifically with regards to the word me is that if you pronounce it with an EE, it doesn't sound very aggressive. James Hetfield pronounces this word with E because that's his personal badass style and Avril Lavigne pronounces it as EI because she's raising her voice to a crescendo. In both cases, I think an EE would sound too 'weak'. If you try shouting the word me!, you might find that the vowel sound naturally distorts.
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