Should moderator variables be always a multiplication of two main predictors?

While studying moderation, I have always seen that a moderator is a multiplication of two main predictors. In the below example, $x_1*x_2$ is a moderator.

$Y=b_0 + b_1x_1 + b_2x_2 + b_3(x_1*x_2) + e$

I wonder if the moderator variables can have a form other than multiplication of two main predictors. If they can, showing some examples would be great.

From Robins & Greenland (1992) and Pearl (2001), we also have the idea of natural direct effects ($NDE$) and natural indirect effects ($NIE$), which add (not multiply) to the total effect ($TE$).
Using the variables you defined above — where Y is our outcome, $X_1$ is our (I'll assume binary) treatment, and $M$ is our mediator (for notational simplicity instead of $X_2$) — we have potential outcomes $Y(x_1,\ M(x_1))$. These are the outcomes we'd observe if both the treatment is $x_1$ ($0$ or $1$) and the mediator is what we'd observe when the treatment is $x_1$. So then we define
$$TE = NIE + NDE,$$ with $$TE = E\{Y(1,\ M(1))\} - E\{Y(0,\ M(0))\}$$ $$NIE = E\{Y(1,\ M(1))\} - E\{Y(1,\ M(0))\}$$ $$NDE = E\{Y(1,\ M(0))\} - E\{Y(0,\ M(0))\}.$$