> Partial function is a standard mathematical term, not a programming language (PL) one

Like many things in mathematics, the term has been borrowed and changed. Look, for example at "function". Do you think "function" in programming (even in Haskell) means the exact same thing as in mathematics?

In general, partial functions mean that for some values, a function is undefined. But in a PL nothing is undefined. In some cases there would be an exception; in others the subroutine would loop forever. But a language like Haskell, because it wants its "functions" to resemble mathematical functions, declares that those perfectly normal computational behaviors correspond to "undefined". In imperative languages, where subroutines don't try to correspond to (partial) functions, throwing an exception does not correspond to undefined.