Then you disambiguate by writing psi(k=0). As a card-carrying mathematician I was as appalled as you at first, but I have to say it creates a lot less issues than what you’d think, and now I find myself using this kind of shortcuts when writing on the board or in notes (not in papers). In math there’s a lot of emphasis on the fact that mathematical writing should be compilable, but that sometimes leads to register spill issues where you’re forced to use symbols that don’t really match their use, or even out-of-memory errors (the “oh no I’m out of latin and greek, I’ll now use hebrew” syndrome). In physics since you’re using the same name for everything, you can just use a limited number of registers that have a clear purpose, and never run out of them.
Another quirk of the “names don’t matter as long as it’s correct” motto is that you sometimes find definitions and formulas that are physically strange. For instance, the definition of Fourier transforms of distributions usually goes by extending the first formula in Fourier transform - Wikipedia, which is just weird (is x real space or reciprocal space?). (This particular one is just a bad choice of definition imo, much better to introduce conjugates and define it through good old parseval.)