It hasn't disappeared. The first equation can be expressed in the form of the second by susbsituting:
h(x, y, u) = f(x, y) - c(x, y) u
In a semi-linear equation there is no constraint that the function "h" needs to be linear in u.
Semi Linear PDE
A book book gives a linear first order PDE in the form
My lectures notes has a 'semi linear' PDE in the form
The presence of u on the RHS makes the PDE semi linear however where does the independant variable u itself disappear in the second equation?