Alrighty, some of my strings were buzzing a little, so I decided to raise the action.
Since I play a Fender Stratocaster, I reached for this ultra-tiny allen wrench and
tweaked those super small screws at the bridge, a pair for each string.

I have successfully eliminated the buzz, however now all of my strings have
their own action, i.e. each string is not on the same height from the fretboard.
Is this normal and common place thing or should all the strings have the same action?

Then goes another thing, which is that the intonation changes after almost every
tweak, action included (at least that's what I read), so I decided to check if intonation
is right. For this, I picked each string fretted at the 12th fret and compared
it's pitch with the natural harmonic at the same (12th) fret.
They matched without further tweaking.
Have I done everything right?

P.S. How do I measure the actual action of my strings to see whether or not
my action is still considered low? That being said, what is considered a low action?

Thank you very much for you time.

Well I did the same thing and got like the action different at each string. It slows you down a lot and is a big hastle. Cause if your picking from string to string you got to like conquer mountains. If you get used to it then thats fine but its gonna slow you down.
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The low E string should be higher off the fretboard than the high E string. I think 1.2 or 1.4 mm should be the min. height above the fretboard of low E at the 12th fret.
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