Actually Bourbaki proved that, in general, from $H_1(\underline{\bf x},M)=0$ one can obtain that the sequence $\underline{\bf x}$ is $M$-quasi-regular. Now, if we want to be $M$-regular need to impose extra-conditions, the most common being $R$ noetherian, $M$ finite and the sequence contained in the Jacobson radical.

@Manos Sorry, I've thought that $I=(x(y-1),z(y-1))$. Please read my comment using this notation.

+1. Your example was the first that crossed my mind too (as a counterexample to Manos' question) since it is known as having a bad behaviour with respect to grade, that is, grade of $I$ is 1 while the grade of $I+yR$ is 3.

Matsumura's Corollary is wrong and the example given by @Youngsu shows this pretty clear.

It would be nice to know who introduced this terminology and when (Rees, Serre, Auslander-Buchsbaum, or earlier)?

@QiL'8 Maybe I'm missing something: in your case, if, for instance, $r=1$ and $R/(x)$ is regular, then the embedding dimension of $R/(x)$ equals $\dim R/(x)=\dim R-1$. I think now it follows that $x\notin\mathfrak m^2$. (What I want to say is that in the end the elements $x_i$ turn out to be outside of $\mathfrak m^2$, although you are not assuming this.)