When I read the notes, a convolution is defined as:
$(f*g)(x) =\int_{-\infty}^{+\infty} f(\tau)g(x-\tau)\rm{d}\tau.$
What is the difference if we define a convolution integral as follows:
$(f*g)(x) =\int_{-\infty}^{+\infty} f(\tau)g(\tau-x)\rm{d}\tau.$
The second would be $(f\ast h)(x)$ where $h(x)=g(-x)$. So unless $g$ is symmetric around $0$, they would not be the same function.
As to the reason we use the first definition rather than the second, as Henry commented, $f\ast g(x)$ is $\int_{x=\tau+\nu} f(\tau)g(\nu)\mathsf d \tau=\int_\Bbb R f(\tau)g(x-\tau)\mathsf d \tau$