However you tweak the other notes (like, in E9th, how many cents flat you tune them) the "E"s have got to be right on - usually the "B"s too. The other notes get "sweetened" to fit in with the "E"s. Because it's an "E" based tuning, all the other strings are tuned from that. So when your guys re-tuned to
your "E", it came good.
When you put your bar on the first fret, then it becomes an "F" based tuning but, because you tuned the
intervals of the open strings (and the pedals) from one string, all the
intervals between strings remain the same, just worked off an "F" note. And so on up the fretboard - F#, G, G#, A etc. etc. so every chord on any fret is ok because the
intervals are relative to the same originally tuned string.