@ brainrainbow: Welcome to the forum.

Being that your E string is a solid steel string, (more likely than not) and the fact that you are plucking to tune, think of your finger as more or less a mute. You're not getting the crisp clean "meant to be" sound that you will get with the bow. By plucking the string you are not getting the same amount of vibration's because a pluck is just that, a pluck and not a sustained tone. Also, depending where you are plucking the string, most people when they  pluck to tune will pluck somewhere half way between bridge and nut. The placement of the tuner also has some effect on tuning. By plucking the string, the string is not vibrating long enough for the tuner to zero in on the exact hz. If I "pluck to tune", I usually rest the end pin of the violin on my knee which can also produce a muted effect. 

I use an online tuner, but not an etuner and use my ear for tuning. The other day I was sitting watching TV and decided to try an online tuner using my iphone. I was amazed that the closer I got to exact pitch I could hear the change in sound coming from my violin. It began to "wobble" If that's a good word for it, it was like a slow vibrato, I couldn't believe it. I got more of the "wobble" the closer I got to being in tune.   P.S. I did mute the TV while tuning.

Don't know if I've given a valid explanation or not, but my advice is to tune using the bow, after all, that's what you'll be using while playing. 

I've also noticed that when I use a concert mute on the bridge I have to be dead on the note played, otherwise I can really hear if my intonation is even a slight bit off.

I hope that give's some kind of insight to the E string mystery, LOL

Good luck.

Ken.