At your link to MathWorld, "linearization" is to be understood as a Fréchet derivative of the appropriate nonlinear mapping, which has become quite habitual nowadays. Your second-order PDE is quasilinear, i.e., linear w.r.t. the highest-order derivatives. A strict formal definition of a quasilinear nonlinear PDE is generally being omitted mostly due to its rather awkward styling that can hardly be avoided. Indeed, in your case, the strict formal definition could look like this. First, we describe $a,b,c,d,e\,$ as functions of $x,y,u$, with $u$ being some unknown function of independent variables $x,y$. Second, we introduce a function $F=F(x,y,p,q_1,q_2,r_{11},r_{12},r_{22})\,$ of eight independent variables being a linear polynomial in its last three variables $\,r_{11},r_{12},r_{22}\,$ reserved for the highest-order deivatives. Namely, in your case, it is
$$
F=a(x,y,p)r_{11}+b(x,y,p)r_{12}+c(x,y,p)r_{22}+d(x,y,p)(q_{1})^2+e(x,y,p)q_{2}\,.
$$
And finally third, we proclaim your equation to be of the form
$$
F\bigl(x,y,u(x,y),u_x(x,y),u_y(x,y),u_{xx}(x,y),u_{xy}(x,y),u_{yy}(x,y)\bigr)=0\quad \forall\, x,y.
$$
At any point $(x,y)$, classification of your equation is determined by the value of the expression
$$
D(x,y)\overset{\rm def}{=}\Bigl(b\bigl(x,y,u(x,y)\bigr)\Bigr)^2
-4a\bigl(x,y,u(x,y)\bigr)\!\cdot\! c\bigl(x,y,u(x,y)\bigr),
$$
with the equation being called elliptic whenever $D(x,y)<0$, hyperbolic whenever $D(x,y)>0$, and parabolic whenever $D(x,y)=0$. There can be absolutely nothing else to it, though one can't help making a remark that identifying the parabolic type with just $D(x,y)=0$ now amounts to a thorough anachronism still tolerated in PDE as something like Historical Landmark.
Fully nonlinear PDE. In case a nonlinear PDE is not quasilinear,
classification is made judging by the linear part of the nonlinear mapping,
i.e., by its Fréchet derivative that dominates questions of
local solvability for the nonlinear mpapping. Just to illustrate how it works,
consider some simple example of the second-order nonlinear partial
differential operator, say,
$$
L(u)\overset{\rm def}{=}F(u_{xx},u_{xy},u_{yy}),
\quad F\in C^1(\mathbb{R}^3),
$$
defined on differentiable functions $u=u(x,y)$ with some suitable choice of
function spaces. The Fréchet derivative of nonlinear mapping $L$ at
the solution $u$ is a linear partial differential operator with variable
coefficients
$$
L_u(v)\overset{\rm def}{=}F_p(u_{xx},u_{xy},u_{yy})v_{xx}+
F_q(u_{xx},u_{xy},u_{yy})v_{xy}+F_r(u_{xx},u_{xy},u_{yy})v_{yy}
$$
where notations $F_p\,,F_q\,,F_r$ are meant to signify the partial derivatives
$$
F_p={\partial_p}F(p,q,r),\quad F_q={\partial_q}F(p,q,r),
\quad F_r={\partial_r}F(p,q,r).
$$
At any point $(x,y)$, classification of the linear partial differential
operator $L_u(v)$ is of course determined by the value of the expression
$$
D_u(x,y)\overset{\rm def}{=}\bigl(F_q(u_{xx},u_{xy},u_{yy})\bigr)^2
-4F_p(u_{xx},u_{xy},u_{yy})\!\cdot\!F_r(u_{xx},u_{xy},u_{yy}).
$$
Hence, the nonlinear partial differential operator $L(u)$ at the solution $u$
is called elliptic at a point $(x,y)$ whenever $D_u(x,y)<0$, and likewise so on.
