Two rings...are they isomorphic? Edit:  I have reverted my question to its original version (which Bjorn Pooenen answered correctly) as requested in the comments.
Consider the local rings
$$R = \mathbb{C}[[x,y,z]]/\langle xy+xz+yz\rangle$$
and
$$S = \mathbb{C}[[x,y,z]]/\langle xy+xz+yz+xyz\rangle.$$
Is $R$ isomorphic to $S$?
Some context:  I am trying to understand formal neighborhoods of points on certain varieties.  I expect one answer, and I'm getting a different answer.  This is the first nontrivial case where the answer that I get does not obviously agree with the answer that I expect.
 A: Yes, they are isomorphic.
More generally, if $k$ is any algebraically closed field of characteristic not $2$, and $n$ is given, then all $k$-algebras of the form $k[[x_1,\ldots,x_n]]/(f_2+f_3+\cdots)$, where each $f_i$ is homogeneous of degree $i$, and $f_2$ is a nondegenerate quadratic form, are isomorphic.  (I.e., there is only one kind of ordinary double point.)
You can construct an isomorphism to $k[[x_1,\ldots,x_n]]/(x_1^2+\cdots+x_n^2)$ with your bare hands as follows.  First diagonalize the quadratic form to assume that $f_2=x_1^2+\cdots+x_n^2$.  Let $m$ be the lowest degree monomial of degree greater than $2$.  Then $m$ is divisible by some $x_i$, say $m=x_1 g$.  Performing the analytic change of variable $x_1 \mapsto x_1-g/2$ eliminates $m$ at the expense of introducing new terms of even higher degree.  By iterating, one can eventually eliminate all monomials of degree 3 to obtain $f_3=0$, and then $f_4=0$, etc.  The partial compositions of this sequence of analytic coordinate changes converge to a single analytic coordinate change because they stabilize modulo any given power of the maximal ideal.
