They are extremely different states of being.

You are happy to receive good news, or for something to turn out well, or whatever else. But it is not a resting state. It's a liminal state. Contentedness, by contrast, is a resting state. You can awake contented, fall asleep contented, and spend your days contented. You may rarely, if ever, experience happiness - yet find yourself able to find satisfaction in life nonetheless.

By contrast a pitiful, depressed, self loathing individual, can experience happiness as much as anybody else. But he is most certainly not content nor satisfied. Perhaps a junky would be another example. A junky certainly experiences happiness when his poison enters his veins, yet he almost certainly is far from content or satisfied.

You are just being pedant while answering to a non native english speaker.

No, the words just have extremely different meanings. A child opens a Christmas present and starts rejoicing - nobody would claim 'Ah, look at him - he's so content!' One could even take this a step further to add that children, in general, cannot be content.

Contentedness is not happiness. And happiness is most certainly not contentedness. They're are just entirely different states of being.

First, it's important to understand that words mean what people tend to use them to mean.

Second, "happy" and "happiness" subsume numerous meanings. You're picking one, and as I and others have, at times, done with many other words, trying to restrict the world to only that meaning.

Perhaps a more similar word to your meaning would be "joy"? It seems more generally restricted to descriptions of brief periods, in common usage.

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.