I'm confused why the Hamilton Jacobi Bellman equation: $$\frac{\partial u}{\partial t}(t,x)+\Delta u(t,x) -\lambda||\nabla u(t,x) ||^{2}=0$$
is considered fully nonlinear, but not semilinear.
By definition, fully nonlinear means the equation is nonlinear in its highest-order terms. But the highest-order terms for this equation are in the $\Delta u(x,t) = \sum_{i=1}^{n} \frac{\partial ^2u}{x_{i}^2}$ , which are linear.
A semi-linear PDE reads $ \mathcal Lu=F(u), $ where $\mathcal L$ is a linear operator and $F$ is a function.
A quasi-linear PDE with order $m$ reads $ \mathcal L\bigl((\partial_x^\alpha u)_{\vert \alpha\vert\le m-1}\bigr) u=F\bigl((\partial_x^\alpha u)_{\vert \alpha\vert\le m-1}\bigr) $ where $\mathcal L\bigl((\partial_x^\alpha u)_{\vert \alpha\vert\le m-1}\bigr) $ is a linear operator of order $m$ whose coefficients are functions of $(\partial_x^\alpha u)_{\vert \alpha\vert\le m-1}$ and $F$ is a function.
Your equation is not semi-linear, but quasi-linear.
In my experience, fully nonlinear refers to an equation of the form $F(x,Du,D^2u)=0$ such that the linearization at $u=u_0$ is a linear equation of the required type (elliptic, hyperbolic etc.). In the example you mention I think the standard would be to call it a quasi-linear equation, and I am quite sure some people would call it semilinear since the nonlinear term does not depend on second order derivatives. Let me add that there is no universally accepted terminology for this, and in many cases the description depends on the problem (and its history).
