Here is a (hopefully correct) elementary proof of the minimality of the generating set. Suppose $I = (f,g)$. Then $I = (f + rg,g)$ for any polynomial $r$, and so we may assume that $f$ and $g$ do not have the monomial term $z^2$ in common. Also, note that since $f,g \in I$, the following terms cannot be nonzero in $f,g: 1,x,y,z,xy,x^2.$ Since we must have polynomials $a,b$ such that $z^2 - x^2y = af + bg$, somehow a nonzero $z^2$ must appear. It cannot come about from a $1$ or $z$ in $f,g$, and so one of these must contain the term $z^2$. WLOG $f$ does, and so $a$ must have nonzero constant term.
Now choose polynomials $p,q,r$ with $f = p(z^2 - x^2y) + q(x^3 - yz) + r(y^2 - xz)$. Since $z^2$ appears in $f$ as a nonzero term, the constant term of $p$ must be nonzero, and the $-x^2y$ term from $p(z^2 - x^2y)$ cannot be cancelled from any other term from the RHS of the equation for $f$ above. Hence $x^2y$ is a nonzero term of $f$.
Now, write $x^3 - yz = cf + dg$ for some polynomials $c,d$. Then one of the constant terms of $c$ or $d$ must be nonzero and the term $yz$ appear in $f$ or $g$. Suppose that the constant term of $c$ is nonzero. Then the $x^2y$ term of $f$ appears and must be cancelled by some term of $dg$. Hence $g$ must contain one of the following terms: $1,x,y,x^2,xy,x^2y$. The first of these are forbidden since $g$ lies in $I$, while if $g =p'(z^2 - x^2y) + q'(x^3 - yz) + r'(y^2 - xz)$ contains the term $x^2y$, then the constant term of $p'$ must be nonzero, and so $g$ must contain a nonzero $z^2$, contradicting our assumption. So the constant term of $c$ is zero, and the constant term of $d$ is not zero. That case ruled out, we must have $yz$ nonzero in $g$. Note that $x^3$ must also be nonzero in $g$. So the constant term of $q'$ is not zero.
Now write $y^2 - xz = jf + kg$. As before, we know that one of the constant terms of $j,k$ must be nonzero to yield a $y^2$ in the LHS. If the constant term of $j$ is nonzero, then a $z^2$ appears from the $jf$ term and so $1,z,z^2$ must appear in $g$, impossible. So the constant terms of $k,r'$ are nonzero. But then one of $1,x,x^2,x^3$ must appear in $jf$. But the constant term of $j$ is zero, so one of $1,x,x^2$ appear in $f$, impossible.