The first two frets produce a buzzing noise on the high E string. The open high E sounds fine. Also, the second fret sounds similar to the third fret. Where is the problem?
Seems like the third fretwire is slightly high. Tapping it gently may re-seat it, or possibly filing it down, only under the 1st string part. Beware that doing that may move the problem up to the next fret, as the third fret may go too low.
An alternative to sanding the fret is to adjust the string height - with an electric guitar, you should be able to do this at the bridge.
Generally, raising the string would be more of an option if you get buzz on more than a couple of frets.
Your third fret is a little bit high. If you change any part of your guitar it will effects some other parts. So try to do these steps:
1) The best choice is to make 3rd fret lower , it has to be done very carefully , you may effect other bars (if you do it wrong you may have this buzzing on 4th and 5th and etc.)
2) Your second choice is to change the entire string's height with adjusting "bridge". It may effect your playing so don't change it a lot.
3) And if still buzzing! finally try to do something with "nut". It's not recommended and it has to be your last choice. Because it will change other strings's heights.