Most open chords have roots that is not on the low E string, notes like A and C where the roots are on the 5th string or D where the root is on the 4th string. This calls for the low E not to be played in the case of A and C and the low E and 5th for chords like D and Dm.
In order to not play those stings, or hear them when one accidently strum those with the chord, I wrap my thumb over those strings to mute them, a technique that I do not see often in many strumming pick players (maybe because they have better control due to playing guitar for years, which I have not). It doesn't seem to matter if you play the string above the root (the thicker one).
Lets take the A chord for instance. The A string is the root note of the chord. The chord itself consist of an A note, C# note and an E note. Now, if I play the low E string with the chord, the chord still stays an A chord because E is the 5th degree in the chord. There is also no audible difference in the chord when strummed.
This brings me to this question, if I get the low E into a chord like A (or even C as E is also the third degree note in the C chord), does this count as an inversion because the root is not on the base anymore, but either on the 3rd or 5th degree depending on the chord. Also, would it be considered bad practice to play these strings marked x above the root.