The technical reason is that chords are, in their basic form, defined by their lowest note (so an 'Am' should have an 'A' as its lowest note...) the open 'E', although in the triad, robs the 'A' of its status; the chord becomes 'E, A, C' instead of 'A, C, E', and could be named 'E sus4#5', following the 'lowest note gives the name' notion. In practise, everyone would still call it 'Am', or better still 'Am/E' (the stroke indicates that it has an 'E' as a bass note...).
I'd echo the above ^^ advice about strumming all the strings, all the time, and try to only play the strings that 'count'; in the 'Am' example, the low 'E' would, indeed, 'muddy' the sound.
All chords have multiple names, for the same set of notes, and it should be the lowest note that gives the 'usual' name. These 'enharmonic' chord names are seldom used, though, except in particular circumstances when composing specific harmonies. One may play an instrument for one's whole life and never come across this; they do occur in orchestral composition, or jazz themes, though; here's one from the Mickey Baker jazz method...
It takes its name from the 'root' 'G' on the 'D' string, or from the 'missing' root 'E' played by a bass player, and absent from the chord on guitar. Either way, only the strings fingered are played, never the 'open' strings.
Disclaimer : I tend to prefer other voicings, and would privilege 'moveable' chords, that don't use open strings at all, and can therefore be played anywhere on the neck. I seldom play all six strings (usually four, often three...) by 'clawing' with my right-hand fingers, rather than strumming 'up'n'down'. There are exceptions (jig'n'reels in 'D', using hammer-on's for melody lines, or open tunings...).
Hope this helps.