I feel that "deterministic" is probably a better word here than "idempotent".
For example, let’s consider a function that accepts a string as an argument and then writes that string to disk. We can consider the disk state as a side effect of the function.
The function itself is perfectly deterministic (output string is a predictable and consistent function of input string), but depending on the implementation of side effects it may not be idempotent. If, for example, this function room simply added the output to a file “output.txt”, this file would grow with every incantation, which is not idempotent. If instead we overwrote the output file so that it reflects only the singular output of the previous run, then the side effects would also be deterministic, that would be idempotent.
At a pedantic level you could redefine your scope of deterministic to not just include outputs, but also include the external state and side effects, but for practical purposes the above distinction is generally how deterministic and idempotent would be applied in practice in computing. I cannot speak to the math-centric view, if there is a different definition there.
But that simply means it's harder to implement true idempotency when it comes to disk usage.
This is why the problem is usually simplified to ignore unhappy paths.
The idempotent version of this function doesn't blindly write. Most of Ansible, for example, is Python code doing 'state machines' or whatever - checking if changes are needed and facilitating if so.
Where y'all assume one function is, actually, an entire script/library. Perhaps I'm fooled by party tricks?
Beyond all of this, the disk full example is nebulous. On Linux (at least)... if you open an existing file for writing, the space it is using is reserved. Writing the same data back out should always be possible... just kind of silly.
It brings about the risk of corruption in transit; connectors faulty or what-have-we. Physical failure like this isn't something we can expect software to handle/resolve IMO. Wargames and gremlins combined!
To truly tie all this together I think we have to consider atomicity
Depends on the implementation: maybe you open a temp file and then mv it into place after you are done writing (for increased atomicity)?
But as I already said, in practice we ignore these externalities because it makes the problem a lot harder for minor improvements — not because this isn't something software can "handle/resolve".
This is deterministic (doesn't change randomly), but not idempotent.