Aranka usually turned into Goldie in English (since that is literally what it means), which I would not consider a name change. Turning it into Gladys (which may be from an old Welsh name that's commonly associated with Claudia) may be a sign of the same line of thinking that leads to the modern American Jewish custom of honoring deceased ancestors with names that share a first initial. Or (or also), Goldie and Gladys share G-L-D in that order, so perhaps someone who didn't (yet) speak English chose the wrong "equivalent", and then it stuck?
In any case, I don't think there was any sort of extra documentation for this kind of change. You could (and as far as I know, still can) choose a totally new name at naturalization; you just had to provide the name you arrived under at the start of the process. In that sense, the naturalization _is_ the documentation of the name change.