A system of parameters need not minimally generate the maximal ideal

The quotient $R\mathrm{/}(\bar{x})\cong k[y{]}_{(y)}\mathrm{/}(y^2)$ is Artinian local, so $(\bar{x})$ has radical $\mathfrak{m}$. Thus there exists a one-generated ideal with maximal radical, and [L2] gives $\dim R\le 1$.

If $\mathfrak{m}$ were principal, then $\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2$ would be one-dimensional over the residue field. But the classes of $\bar{x}$ and $\bar{y}$ are linearly independent modulo ${\mathfrak{m}}^2$, so $\mathfrak{m}$ is not principal. In particular $\mathfrak{m}\mathrm{\ne }(0)$, so $R$ is not a field.

Since $R$ is a local domain and not a field, [L3] shows that $\dim R\mathrm{\ne }0$. Together with step 1.1, this forces $\dim R=1$. Then [L1] makes $(\bar{x})$ a parameter ideal.

Therefore $(\bar{x})$ is a parameter ideal while $\mathfrak{m}$ is not principal, so a system of parameters need not minimally generate the maximal ideal.