Your example model can be reëxpressed to be linear in the parameters $\alpha=\beta_1\beta_2$ & $\zeta=\exp\beta_3$:
$$g(\operatorname{E} Y) = \beta_0 + \alpha x_1 + \zeta x_2^2$$
(Clearly $\beta_1$ & $\beta_2$ aren't separately estimable; a non-linear model wouldn't help there. And note that $\hat\zeta$ must be constrained to be positive.) Some models can't be so reëxpressed:
$$g(\operatorname{E} Y) = \beta_0 + \beta_1 x_1 + x_2^{\beta_2}$$
Some can be though it's not obvious at first: https://stats.stackexchange.com/a/60504/17230.
There's a very thorough discussion of different meanings of "linear" at How to tell the difference between linear and non-linear regression models?.
