For a finite module, the dimension is the least size of an ideal of definition, and such tuples are systems of parameters

Let $A:=R\mathrm{/}{\mathrm{Ann}}_R(M)$. Then $M$ is a faithful finite $A$-module and ${\mathrm{Supp}}_R(M)$ identifies with $\mathrm{Spec}(A)$, so $d=\dim A$. The maximal ideal of the Noetherian local ring $A$ has finitely many generators by [L3], and [L4] bounds its height by that finite number. Thus $d<\mathrm{\infty }$, so [L1] applies to $A$.

For an ideal $J\subseteq R$, the quotient $M\mathrm{/}JM$ has finite length if and only if $(A\mathrm{/}JA{)}_{\mathfrak{p}}=0$ for every nonmaximal prime $\mathfrak{p}$ of $A$, which is equivalent to $\sqrt{JA}={\mathfrak{m}}_A$. Thus ideals of definition for $M$ are exactly the ideals of $A$ with maximal radical.

Applying [L1] in the local ring $A$, the least number of generators of an ideal with radical ${\mathfrak{m}}_A$ is exactly $\dim A=d$. By algebra this is the least number of generators of an ideal of definition for $M$.

If $x_1,\dots ,x_d\in \mathfrak{m}$ satisfy that $M\mathrm{/}(x_1,\dots ,x_d)M$ has finite length, then algebra gives $\sqrt{(x_1,\dots ,x_d)A}={\mathfrak{m}}_A\mathrm{.}$ Therefore their images in $A$ form a system of parameters by [L2], and we call the original tuple a system of parameters for $M$.

This proves both claims.