**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.