The problem is that there are two common definitions of the impulse response of an LTV system, resulting in the following input-output relations:

$$ y(t)=\int_\tau h_1(t,\tau)x(\tau)d\tau\tag{1}$$

and

$$ y(t)=\int_\tau h_2(t,\tau)x(t-\tau)d\tau\tag{2}$$

In the first one, the impulse response (integration kernel) $h_1(t,\tau)$ is the response at time $t$ to an impulse at time $\tau$. In the second, $h_2(t,\tau)$ is the response at time $t$ to an impulse at time $t-\tau$. The impulse response $h_2(t)$ is also called the *input delay-spread function*.

The relationship between $h_1(t,\tau)$ and $h_2(t,\tau)$ is

$$h_1(t,\tau)=h_2(t,t-\tau),\quad h_2(t,\tau)=h_1(t,t-\tau)\tag{3}$$

The conditions for time-invariance are as follows. For the first definition we require that $h_1(t,\tau)$ only depends on the difference $t-\tau$:

$$h_1(t,\tau)=\tilde{h}(t-\tau)\tag{4}$$

The second definition results in an impulse response that is independent of $t$:

$$h_2(t,\tau)=\tilde{h}(\tau)\tag{4}$$

The causality conditions are $h_1(t,\tau)=0$ for $t<\tau$ and $h_2(t,\tau)=0$ for $\tau<0$.