What is it called if a system is "in rest" until time $t=t_0$?
There are two things to consider here. First of all, a non-causal system can react to input that hasn't arrived yet. And, secondly, there could be non-zero initial conditions that would contribute to the output signal without any input signal.

So what you probably mean is a "causal system initially at rest" or "a causal system with zero initial conditions".