What is the meaning of the half saturation constant in the following system, describing the interaction between a population of predators of size $y$ and a population of preys of size $x$? $$\dot x=x(a-bx)-c\frac{xy}{my+x}\qquad\dot y=-dy+f\frac{xy}{my+x}$$ I am interested in what the parameter $m$ represents. In this article it is mentioned that it is the half saturation constant.
Does it mean the prey density when the predator population is its half maximal capacity?
This is at best an abuse of terminology. For a predator dynamics $$\dot y=-dy+y\frac{fx}{m+x}$$ the specific growth rate of the predator population due to the preys is $$r=\frac{fx}{m+x}$$ which reaches its maximum $r_{\text{max}}=f$ when $x\to\infty$ (that is, at saturation). Then, $r=\frac12r_{\text{max}}$ when $x=m$ (hence the name, half saturation constant).
In the model you are considering, the analogue of $r$ is $$\bar r=\frac{fx}{my+x}$$ hence $\bar r_{\text{max}}=f$ again, and $\bar r=\frac12\bar r_{\text{max}}$ when $x=my$. Thus $m$ would be the half saturation constant in the sense that half saturation (that is, $\bar r$ is at one half of its maximal value) occurs when the ratio $x/y$ is at $m$.
This is the best analogue I could find.
Note anyway that in the version you are considering, $m$ is dimensionless instead of homogenous to $x$ and $y$, hence an interpretation of $m$ as a population such that this or that happens, is impossible.