I would have to question the analysis that it is indeed buzzing on the fret you are playing on - that's a very difficult thing to say with absolute certainty. It's possible if during the leveling the fret was ground very flat on top and not properly recrowned, but unlikely.
The more likely options are these. The string is buzzing on the following fret, and as you move your finger further toward the played fret you are in effect increasing the arc as it extends over that fret, thereby raising the string higher above the following fret. This is one reason for the common capo buzz when they are placed too far back from the fret. If this is the case you would obviously need to focus on your fret leveling.
Another possibility is back buzz, or buzzing on the fret(s) between the preceding fret and the nut. These can be very finicky, and require a very specific set of circumstances. As you move your finger and change the angle the string meets the preceding fret at (again, thus changing the arc and string height beyond that fret) the buzz can disappear as the string gets farther from or closer to the fret. If this were the case you would need to raise the nut, and/or possibly benefit from dressing the frets as well. This is a symptom more commonly found on classicals than steel strings. It usually occurs at frets higher than 2 or 3 though, where the string length behind the fret is longer and more receptive to being driven to sympathetic vibration.
Again, it's possible for a string to rattle sitar-like on a very flat fret, but that would be low on the list of usual suspects. If it were the case, it would be a very different sounding buzz than the common fret rattle, usually being much more faint. Again, it would also require a very, very flattened fret.
_________________ Eschew obfuscation, espouse elucidation.
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