Question regarding localization of polynomial ring

Question

I made an exercise that went as follows: Suppose $R = \frac{\mathbb{R}[x,y]}{(xy)}$. Define the multiplicative set $$ S = \left\{ 1 + (xy), x + (xy), x^2 + (xy), \ldots \right\}.$$ In the exercise, I showed that $S^{-1}R \cong \mathbb{R}[x,x^{-1}].$ My question is as follows: Why don't we just define $R = \mathbb{R}[x]$ and consider the multiplicative set $S = \left\{ 1,x,x^2, \ldots\right\}$? Wouldn't we also then get $S^{-1}R \cong \mathbb{R}[x,x^{-1}]$? If so, what is the advantage one gets with the definition used in the exercise, if any?