System Response Terminology If I have a system specified by 

$$P(D)y(t)=Q(D)x(t)$$ 

and I specify initial conditions $y(0^-)=a, \ y'(0^-)=b,\ x(0^-)=c$ does the term $x(0^-)=c$ correspond to the zero state response or zero input response?
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.