Thanks, Tomk80, I think you have got it. If a single photon of radiation was the cause of an error in replication, then you cannot say it has been directly observed unless you saw the photon hit during the process of replication, and knock something askew.
But if that's all you saw, don't call the mutation "random" For that you need to know where it came from. Maybe it was directed by humans following Hiroshima. If so, it might still be random as to which gene the photon hit, kind of like the roulette wheel ball. But if a human directed the photon at a particular gene for the purpose of creating the mutation, then the mutation clearly was not random. I do not know whether we have that ability yet.
Now, the final part is, has the mutation improved the survivability of the species so that we can say it was beneficial. It will take some time, but natural selection will have to be taken into account to determine whether the mutation was beneficial or not.
Now if your observation of whatever does not meet these requirements, you should not say that a "beneficial random mutation has been directly observed."
Some of the requirements can be deleted by deleting the adjectives.
I am not well versed on what causes mutations. I assume that things other than errant radiation photons can do it. But whatever does it, you've got to directly sense it to directly observe it. Don't say you have, if you haven't.