In English there are two possible translations, but the meaning is not quite the same. Does this distinction make sense in German? If so, how would one make the distinction?
"You are happy with life" = You are generally very happy; you are very happy with YOUR life."
"You are happy with THE life" refers more to a way of life in which you participate, not just your own life; for example "You are happy with the life in a monastery" (if one lives in a monastery) or "You are happy with the life of a pirate (including long months at sea, sleeping in a hammock, getting wet often, drinking bad rum)."
"Contented" sounds fine to me (US English native speaker), and Merriam Webster agrees with its use as an adjective. "Contented" is perfectly fine in English. See this: https://www.merriam-webster.com/dictionary/contented
yes, I find those two you mentioned akin to the concept. the content person is happy but perhaps in a peaceful, calm way, or just through satisfaction. happy is more general; you can be not content yet still happy...of wanting more and more [of something]. but you're right anyway, that's what people generally make use of the adjective, sadly.