I think the point Phil tries to make is the difference between “instantaneous utility”, that is a function on things at some point in time (actually, phase space), and the “general utility”, which is a function that also has time (or position in phase space) as an argument.

While not immediately obvious, I think his naming choice could be worse. According to my non-scientific poll of one (me), when seeing the word “happiness” people think of time as a parameter instinctively, but consider specific instants for “utility” unless there are other cues in the context.

A strict definition such as yours would require coining a few new words for the discussion. That's not a bad thing per se, I just can't think of any that have the advantage of being already used as such in general vocabulary.