Using the definition given by wikipedia https://en.wikipedia.org/wiki/Partial_function a partial function $f:X\nrightarrow Y$ is just a function $f:X'\to Y$ for some subset $X'\subseteq X$ analgous to the way for a categorically defined function its image is some subset of its co-domain/target set. However they go on to say a "partial transformation" is a partial function from some set to itself. Yet by that description I could define a partial function $f:X\cup Y\nrightarrow X\cup Y$ analgous to our previous one by just throwing in extra values its not defined at or doesn't map to until I get one that maps from the same set back to itself and yet corresponds to an identical function $f:X'\to Y$ as it did previously.
So am I missing something here? In short it appears a partial function is just a functional relation equipped with two externally defined sets which just have to contain the domain/image. Though since we can always construct external sets equal to each other satisfying this whose associated functions are the same, it appears to me that the distinction between a partial function and a partial transformation is entirely arbitrary. Can anyone clear this up for me? Where did I go wrong?