Very elementary question here, but I just wanted to know when two affine varieties are considered to be equal. To give the most fundamental example I can think of. The varieties $V_1$ and $V_2$ defined by \begin{equation*} V_1 := \{ x \in \mathbb{R} | x = 0 \} \end{equation*} and \begin{equation*} V_2 := \{ x \in \mathbb{R} | x^2 = 0 \} \end{equation*} both consist of the same point, $x=0$. Are these varieties equal because they consist of the same point in $\mathbb{R}$, or are they unequal because they are each defined by different polynomials, $f(x)=x$ and $g(x)=x^2$, respectively?
Specifically, is it the points that define the variety, or is it the polynomials that define the variety?