System Response Terminology
Assuming that you denote the output by $y(t)$ and the input by $x(t)$, the initial conditions are given by the values of $y(t)$ and its derivatives at a certain time $t_0$ (choose $t_0=0^-$ if you like). A value $x(t_0)$ has nothing to do with an initial condition.

For the zero-input response (ZIR) you just set $x(t)=0$ and compute the output caused by possibly non-zero initial conditions. For the zero-state response (ZSR), you set $y^{(k)}(t_0)=0$, $k=0,1,\ldots$, and compute the output for a given input $x(t)$. The total response is then the sum of the ZIR and the ZSR.