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.
Best Answer
Perhaps you should study some more advanced analysis, since that's when Frechet derivatives come up. A good (and legally free) reference is Applied Analysis by John Hunter and Bruno Nachtergaele.
After that, perhaps Analysis by Elliott H. Lieb and Michael Loss? It's more advanced, so be sure you understand Hunter and Nachtergaele first.
For more intense partial differential equations, UC Davis' upper division PDE's courses are available online too (with homework and solutions) when Nordgren taught it. There is Math 118A: Partial Differential Equations and 118B.
There are probably more advanced (free) references out there, but these are the ones I use...
Addendum: For references on specifically functional analysis, perhaps you should be comfortable with Eidelman et al's Functional Analysis. An introduction (Graduate Studies in Mathematics 66; American Mathematical Society, Providence, RI, 2004); I've heard good things about J.B. Conway's Functional Analysis, although I have yet to read it...