That is the first definition stated in first-order logic, sure. But the operational reading of a (computable) function being something that takes in input and p

That is the first definition stated in first-order logic, sure. But the operational reading of a (computable) function being something that takes in input and produces an output (via some algorithm) is still correct. It is possible to be both clear and correct with operational analogies. However, monads and functors don't readily admit an easy operational reading like functions.