From Wikipedia, with emphasis added: The covering dimension of a topological space $X$ is defined to be the minimum value of $n$ such that every finite open cover $\mathcal{A}$ of $X$ admits a finite open cover $\mathcal{B}$ of $X$ which refines $\mathcal{A}$ in which no point is included in more than $n+1$ elements. The dimension of $\mathbb{R}^2$ is not 1 because there are coverings other than the one you defined, e.g., ones by balls or squares that you alluded to, that witness that the dimension is $\ge 2$.
EDIT: Just to clarify, I think that the OP mentally substituted "some" for "every" in the definition.