Sure a functor satisfies more conditions but that doesn't make it less of a function. Heck if you restrict it to mapping a single set of objects or a particular Homset it is a function, you just can't extend this to the whole category if the objects form a proper class (which happens fairly often).

The whole distinction between value-level or type-level is meaningless in general categories. Sure there's a distinction between morphisms between objects and functors between categories but you've also got the category of categories where the morphisms are functors between categories.

Edit: I'll have to concede that the article confuses them too much though, you can't just pick a random function and declare it to be a functor, especially when it's not clear between which categories.