There is no contradiction but Textbook description is clearer: consider only highest order derivatives. If expression is not linear with respect to them, we get (completely) non-linear; if it is linear with respect to them albeit coefficients depend on lower order derivateeves, it is quasilinear. If coefficients at highest order derivatives do not depend on solution it is semilinear.

**Example**

$a(x,y)u_{xx}+ 2b(x,y)u_{xy}+c(x,y)u_{yy} + d(x,y) u_x + e(x,y)u_y + f(x,y)u=F(x,y)$ is linear.

$a(x,y)u_{xx}+ 2b(x,y)u_{xy}+c(x,y)u_{yy} =F(x,y,u,u_x,u_y)$ is semilinear (unless $F$ is linear with respect to $(u,u_x,u_y)$ plus $g(x,y)$ in which case we are in the previous framework).

$a(x,y,u,u_x,u_y)u_{xx}+ 2b(x,y,u,u_x,u_y)u_{xy}+c(x,y,u,u_x,u_y)u_{yy} =F(x,y,u,u_x,u_y)$ is quasilinear.

$F(x,y,u,u_x,u_y,u_{xx},u_{xy},u_{yy})=0$ is nonlinear.