I'll give this one a go. I understand it this way. Whether Major or Minor, chords are understood in light of the Major scale. A Major chord features the 1, 3, and 5 of the scale. The minor chord of the same key also has the 1, 3, and 5--with the tweak of flattening the 3rd. So, an E6 features an E Major chord (1-3-5, or E-G#-B), plus the 6th of the E Major scale (C#), making an E6 look like (E-G#-B-C#). The Em6 likewise features the (1-3-5, with the flattened 3rd, so it's really 1-b3-5); and again, the 6th of the E Major scale is added (C#), making an Em6 look like (E-G-B-C#). In his book, Guitar Chords, Paul Roland offers three diagrams each for the E6 and Em6 chords, and each set features the same 4 notes for each chord. Michael Wolfsohn, in his book, Ultimate Guitar Chord User's Guide, the 5th of both E6 and Em6 are listed as "preferred notes." Of course, the presence of the 6th gives the chord its distinctive name, and in fact both the E6 and Em6 require the 1, (b)3, and 6. From the look of things, if the chord is easier played with or without the 5, then perhaps that's the way to play it. If you like the sound of one or the other in your chord progression, and you can play the preferred Em6 without much difficulty, then go for it. Peace!

The "m" in "Em6" refers to the 3rd, not the 6th. So a minor 6th chord contains a root, minor 3rd and a 6th. The chord that contains a root, major 3rd and a flattened 6th (usually referred to as a sharpened 5th in this context) would be called an augmented chord.