Generators of $\mathbb{Z}[x]$ ring.
Solution 1:
I think your confusion is that you are interpreting "1 and $x$ are generators of $\mathbb{Z}[x]$" as "the set $\{1\}$ is a generating set of $\mathbb{Z}[x]$, and the set $\{x\}$ is also a generating set of $\mathbb{Z}[x]$". That statement is untrue. The correct inrepretation is "the set $\{1,x\}$ is a generating set of $\mathbb{Z}[x]$.
(in fact $\{x\}$ is not a generating set either, since $x^{-1}$ is not included in $\mathbb{Z}[x]$).
Similarly, saying that $1$, $x$, and $y$ are generators of $\mathbb{Z}[x,y]$ means that $\{1,x,y\}$ is a generating set.