Which is better wording: x grows with $N^2$ or x grows with $O(N^2)$
The three options have different meanings in mathematics:


The first option means that $x$ increases as $N^2$ increases.
The second doesn't really make sense.
The third has the meaning you are presumably after (perhaps "with" is better replaced by "in").


You can rephrase the first two options as follows:


$x$ grows as $N^2$ - this is informal, and doesn't differentiate between $x \sim N^2$ and $x = \Theta(N^2)$.
$x$ grows as $O(N^2)$ - this is an awkward phrasing, and it is better to use symbols: $x = O(N^2)$. Note, however, that this is not the same as your third option - it corresponds to $x$ growing at most quadratically in $N$. To get the same meaning, you have to use big Theta: $x = \Theta(N^2)$.


You mention rigor. Rigor is a property of proofs, whereas here there are no proofs. Your descriptions might be informal, but could help explain rigorous proofs.

The best two options are "$x$ grows quadratically in $N$" and $x = \Theta(N^2)$". Both are acceptable.