regular local regular quotient ideal is parameter generated

Put $c={\dim }_k((I+{\mathfrak{m}}^2)\mathrm{/}{\mathfrak{m}}^2)$. The cotangent space of $R\mathrm{/}I$ is $\mathfrak{m}\mathrm{/}(I+{\mathfrak{m}}^2)$ and has dimension $d-c$. Consequently the numerical equality is precisely the definition of regularity of $R\mathrm{/}I$.

If $R\mathrm{/}I$ is regular, choose $x_1,\dots ,x_c\in I$ lifting a basis of that subspace and extend their classes to a cotangent basis of $R$. Put $S=R\mathrm{/}(x_1,\dots ,x_c)$. Repeated parameter reduction makes $S$ regular of dimension $d-c$, and the extended tuple is a regular system.

The ring $S$ is a domain. If the kernel $J$ of $S\to R\mathrm{/}I$ were nonzero, any prime chain in $S\mathrm{/}J$ would lift to a chain of nonzero primes of $S$, to which $(0)$ can be prepended. Hence $\dim (S\mathrm{/}J)\le \dim S-1$, contradicting equality of dimensions. Thus $I=(x_1,\dots ,x_c)$. Conversely, repeated parameter reduction makes every such quotient regular. This includes $c=0$, when $I=0$, and $c=d$, when $I=\mathfrak{m}$ and the quotient is $k$.