When you swap the G for the E, you get rid of a perfect 5th and get a major 3rd. That 5th is being played anyway on fret 5, D string. You end up with:
- Minor 3rd
- Major 3rd
- Perfect 5th
If you want to see it strictly as a minor C chord, the E is a major 10th tension. This tension is not diatonic to any minor mode from the major scale, so it is an "outside" tension.
It all depends on the harmonic context, but you can also see this chord as a Dominant #9 chord with the 7th ommited. The minor 3rd can be seen as an augmented 9th (#9). That would be a C7#9. Since your chord doesn't have the minor 7th, maybe Cadd#9 would be more appropriate. You could also play the Bb in fret 3 of string G, although a more common fingering would be X3234X, the "hendrix" chord, which ommits the 5th. You should try this one out. Pretty common in jazz as well.
Dominant chords are built on the 5th degree of the major scale. They are formed by a major triad plus a minor 7th chord tone. A C7 belongs to the key of F major, with the notes C, E, G and Bb. That #9 is a D#, an altered tension that doesn't belong to the F major scale, but it's widely used to add dissonance to the dominant chord, even though it's not diatonic to the major scale.
- 1 - Root, C
- 3 - Major 3rd, E
- 5 - Perfect 5th, G
- 7 - Minor 7th, Bb (you left this one out)
- #9 - Augmented 9th, D# (not diatonic to F major, look into altered harmony)
If this didn't help at all, I suggest that you take a look into basic harmony, especially the diatonic 7th chord sequence of the major scale. Once you understand it, study altered harmony.