Initial rest condition applied on $x(t)$ vs $h(t)$
It should be clear that a property of a system, such as causality, cannot be determined by looking at its input signals. For a linear time-invariant system, it is its impulse response $h(t)$ from which properties such as causality or stability can be determined.
Only the second definition in the question is correct: a causal LTI system has an impulse response $h(t)$ which equals zero for $t&lt;0$. This implies that the system's output at time $t_0$ only depends on values of the input signal $x(t)$ for $t\le t_0$, and not on future values $x(t)$, $t&gt;t_0$.
The output signal of a causal LTI system can be written as
$$y(t)=\int_{0}^{\infty}h(\tau)x(t-\tau)d\tau=\int_{-\infty}^tx(\tau)h(t-\tau)d\tau$$
An initial-rest condition just means that if the input $x(t)$ is zero for $t&lt;t_0$ then the output $y(t)$ must also be zero for $t&lt;t_0$.