But it's also quite common to call a "utility function" something that shows/tells/calculates how much you value something specific.

I'm not familiar with that usage, Could you point me to a case in which the term was used, that way? Naively, if I saw that phrasing I would most likely consider it akin to a mathematical "abuse of notation", where it actually referred to "the utility of the world in which <X> exists over the otherwise-identical world in which <X> did not exist", but where the subtleties are not relevant to the example at hand and are taken as understood.

I am not sure of that. Utility functions often map to ranks, for example.

Could you provide an example of this also? In the cases where someone specifies the output of a utility function, I've always seen it be real or rational numbers. (Intuitively worldstates should be finite, like the universe, and therefore map to the rationals rather than reals, but this isn't important.)