If I were simply restricting the meaning, then the examples I am offering would mean you could use happy or e.g. contentedness interchangeably, or at least doing so wouldn't sound ridiculous. Yet I'm sure you're realizing by now that you cannot! For another example - a stern and disciplined person, who is rather dour of character, could be completely contented, but it would be illogical to call this person "happy", however you might want to define the term.

The words are simply not synonyms, or even particularly close to being synonyms.