Why can Laplace’s Equation have a Green’s Function?

Question

On the Wiki page for Green’s Functions, it states that a Green’s Function is an impulse response to an inhomogeneous source term of a differential equation. But, then I read elsewhere that the kernel for Laplace’s Equation is $$G(x;y)=-\frac{1}{4\pi |x-y|}.$$ But, how can this be if Laplace’s Equation is homogenous? There can be no response. Finding the kernel is simple enough, but I’m struggling to explain this inconsistency.