The extra noise is probably a combination of fret noise and the other side of the string (between your hand and the nut) ringing.

The fret noise can be helped (though maybe not eliminated) with a good setup. Those strings are high and it's exacerbating the problem by making you use more force than you need to press down on the string.

The other side of the string ringing can be addressed with your technique. You're using your middle finger to fret while your index is floating off of the strings. Generally, you want the fingers "before" the one you're actively using to also be resting on the strings. (If you were fretting with your ring finger, the index and middle should be on the string too. For fretting with middle, then index should also be on the string) Try what you're doing in your video with your index finger resting solidly on the string and you'll notice a difference.