Yes, it’s a different result.
The people who created variance swaps decided that they would write “20” to mean 20% (= 0.2) in their formula.
If they had decided to write “0.2” to mean 20%, then their formula would have been different (e.g., variance\ notional = \frac{vega\ notional}{0.02×strike\ price}), so that the amount comes out to be the same.
The point is that for the variance swap described in that question, the answer is supposed to be $129,375. You can get that number by changing how you write percentages and how you compute the variance notional, as long as you change them together.
In a way, it’s similar to how convexity is computed and used for bond pricing. They used to have a “2” in the denominator of the formula for calculating convexity, but not in the formula for using that calculated number to price a bond. More recently, they decided to drop the “2” from the formula for calculating convexity, but include a factor of “½” in the formula to use that convexity. The upshot is that the mount of convexity adjustment is the same either way, there are simply two routes one can use to determine it.