That's canon for that Universe, that has nothing to do with whether or not they are different swords in real life.
Except for the fact that it does, because the entire concept of Servants and Noble Phantasms is based on the concept of make-believe things coming to life in forms that are currently accepted by humanity in general, regardless of how accurate they might be, or if said individuals or objects ever actually existed in the first place.
Thus, my argument is quite valid, because the entire mechanic in question is based upon how mythology is perceived. You claim that there are two swords in canon, but that's not entirely true. There are as many as there are different versions of the legend. And they can still all be the same sword, because they're all King Author's sword, and he only ever had one. They're just different versions of the same thing from different branches of Authorian mythology.
It's not the canon number that matters. It's the fact that canonically, you can have two, three, four, or however-many completely different versions of what are essentially the same thing.
It's also worth noting that this is exactly the sort of thing that could happen, because as you yourself pointed out, King Author's sword has two different origin stories, both of which are widely known. Thus, it should not be the least bit surprising that there are two different versions of it floating around.