The problem is that there are two different common definitions of the impulse response of an LTV system, resulting in the following input-output relations:
$$ y(t)=\int_\tau h(t,\tau)x(\tau)d\tau\tag{1}$$
and
$$ y(t)=\int_\tau h(t,\tau)x(t-\tau)d\tau\tag{2}$$
In the first one, the impulse response $h(t,\tau)$ is the response at time $t$ to an impulse at time $\tau$. In the second, $h(t,\tau)$ is the response at time $t$ to an impulse at time $t-\tau$. The impulse response in the second definition is also called the input delay-spread function.
The conditions for time-invariance are as follows. For the first definition we require that $h(t,\tau)$ only depends on the difference $t-\tau$:
$$h(t,\tau)=\tilde{h}(t-\tau)\tag{3}$$
The second definition results in an impulse response that is independent of $t$:
$$h(t,\tau)=\tilde{h}(\tau)\tag{4}$$