I don't think the argument here is only, or even mostly, about the definition of "random." It's also rather fundamentally about the definition of "outcome."
The point of my analogies with air pressure, fission chain reactions, and so forth is that whether you see the "outcome" of a set of randomly decided events as "random" depends on what level of description you look at the system. The movement of air molecules in the tire is random, and which molecules collide with the wall at any given time interval will also be random. But the cumulative phenomenon we call gas pressure is not random. Thus, we can call the outcome random if we care whether Aaron the Air Molecule hits the wall during some specific time interval, but if what we care about is that the rims stay off the pavement, it makes no sense to call the overall phenomenon, gas pressure, random.
Same thing for the chain reaction, which is non-independent events. If the 100th nucleus to fission sends a neutron one way, a certain other nucleus (call it Leon) gets hit and fissions immediately. If it sends it a different way, then Leon might never get fissioned at all by the time the chain reaction is complete. So, the outcome is random if we care about what happens to Leon. But overall, the outcome of the chain reaction is not random, it is the same every time.
An even better analogy might be the Barnsley Chaos Game algorithm. Take four affine transformations of a plane (illustrated
here.) Start at an arbitrary point on the plane. Then, repeatedly choose one of the four transformations at random, and apply it to the previous point. What you get is the familiar "fern" image, shown
here.
Or rather, what you get is a very different set of specific points each time.
There's an uncountably infinite number of points in the attractor, but each running of the algorithm can only collect a countably infinite set of points (and that's only if you run it through infinitely many iterations), so the chance of any specific randomly chosen point being in the set produced by a given run approaches zero. If you don't put any precision limits on your calculations, then each transformation is going to add more nonrepeating decimal digits to the next point's coordinates, which means that any given point in the attractor has at most one chance to be in the set, and only if a unique, or one of a very few, combinations of random choices has been made for a (potentially very large) number of steps.
But, you still get the same picture every time.
Random input to the process. Random outcome, if you care whether Pam, Peter, Paul, Phillip, Polly, or Petunia the Point end up in the set or not. Non-random outcome, if you're looking for a picture of a fern.
So, what do you consider the "outcome" of evolution?
If the outcome is "a large set of specific individual genomes," then yes, the outcome of evolution is random, in the sense of being completely sensitive to and completely controlled by the random inputs from outside the system.
If the outcome is "a world full of diverse and complex life," then no, the outcome is not random. It'll come out that way every time.
Respectfully,
Myriad