Yes, Poisson's equation includes Laplace's equation (with f(x,y)= 0) which is an elliptic equation.
Actually the basic concept of "type" of a partial differential equation is based on the simplified form that you started with. Any equation of the form
Auxx+ Buyy= something is "elliptic", A and B are numbers of the same sign. Any equation of the form Auxx- Buyy= something is "hyperbolic". Any equation of the form Auxx- Buy= something is "parabolic". Having other derivatives just makes things more complicated- but you can always make changes of independent variables to get rid of mixed derivatives and first derivatives. The formula you cite is based on those changes.
The names are, of course, based on the analogy with conic sections.
The distinctions are important because hyperbolic equations always have two "characteristic curves", parabolic equations one, and elliptic equations none.
For example the hyperbolic equation Auxx- Buyy= 0 has "characteristic curves" \sqrt{A}x+ \sqrt{B}y= 0 and \sqrt{A}x- \sqrt{B}y= 0. Setting s= \sqrt{A}x+\sqrt{B}y and t= \sqrt{A}x- \sqrt{B}y reduces the equation to
4\sqrt{AB}u_{st}= 0 (similar to the way x2- y2= 1 can, by a rotation of axes, by reduced to x'y'= 1) which is easy to solve: dividing by the constants, ust= 0. Since the derivative of us with respect to t is 0, us must depend only on s: us= f(s) for any function f. Then integrating again, u(s,t)= F(s)+ G(t) where F(s) is an anti-derivative of f and G is the "constant of integration" which, since this is a partial derivative with respect to s, may depend on t.
The General solution to Auxx- Buyy= 0 is
F(\sqrt{A}x+\sqrt{B}y)+ G(\sqrt{A}x- \sqrt{B}y) where F and G can be any twice-differentiable functions of a single variable.
Since elliptic and parabolic equations don't have two separate characteristic curves, they cannot be done that way.
Finally, if you allow variable coefficients: A(x,y)uxx+ B(x,y)uyy= something, the equation may be "elliptic" for some values of x and y, "parabolic" or "hyperbolic" for others.
