Definition of one-way functions: randomly-chosen point vs every point
The reason you are looking at a random choice of $x$ is that, otherwise, there is no probability space any longer. Put differently, if we require that for every x there should not be an algorithm that given an image $f(x)$ can output a preimage, we have a problem, because there is such an algorithm. For every $x$ consider the algorithm $M_x$ that on any input simply outputs $x$. Thus, we have just defined for every $x$ an algorithm that (on input $f(x)$) inverts the function.