Quasilinear PDE definition?

Yes, that's it. The term $$\frac{\partial u}{\partial y}\frac{\partial ^2u}{\partial y^2}$$ is linear with respect to $\frac{\partial ^2u}{\partial y^2}$, but the term $$\left(\frac{\partial ^2u}{\partial y^2}\right)^2$$ is not linear.

If you have a second order derivative, it does not matter if the equation has a term like $u^2$ to be quasilinear. You just need to check the highest order derivatives (second order ones in the examples of the link).