I certainly do not define it the second way. Most people care about something other than their own happiness, and some people may care about their own happiness very little, not at all, or negatively, I really don't see why a 'happiness function' would be even slightly interesting to decision theorists.

I think I'd want to define a utility function as "what an agent wants to maximise" but I'm not entirely clear how to unpack the word 'want' in that sentence, I will admit I'm somewhat confused.

However, I'm not particularly concerned about my statements being tautological, they were meant to be, since they are arguing against statements that are tautologically false.