1) If you put your finger very close to the fret you want to press, the buzz will lessen. Jaco Pastorius said that he plays fretted and fretless bass the same way, because placing your fingers right on the frets helps deal with the fret noise.
2) You want to learn left as well as right hand muting. Check what's it about here
You can hear that the fret buzz you ask about is in the video too, only much less than in your recording. When you learn to mute, your buzz will lessen, as well as other unwanted noises.
3) You got quite a high action, which makes the fret noise harder to deal with. You should generally focus on using your fingers dexterity, not muscle strenght. I'd put the strings lower. You can do it yourself, it's no rocket science and if you are sensible, you can't do damage that cannot be repaired easily. Read some
fender manuals on how it's done.
4) Don't be worried too much. Most of the small noises will get lost in the mix when you play with a real band; some (actually a lot, style depending) of the noises are even desirable.