While I think that most people in casual dialogue would approximate the meanings of all these words such that they mean pretty much the same thing, I think looking into the etymologies of the words is instructive.
Content: from Latin continere, to contain => contentus, contained — in the context where a person’s desires are bound by what he already has. There was a point in my life where I was astoundingly content in the sense that I was not frustrated by what I lacked, including having possessions, a place of my own to sleep, oodles of money. At the same time I also knew that my standards could evolve, but that contentedness could only be maintained by aligning my emotions to things as they happened in the present.
Satisfied: from Latin “make enough”; related to the ‘sat-‘ in saturate and satiate, where ‘sat’ refers to being full or drenched. I think our formal meaning remains: something is satisfying when it has reached the point of being enough to suit some end. So the state of being satisfied would be resting no longer concerned in the fact that some personal criteria, concern, etc. has been met.
Happy is more related to an delighted emotion or positive reaction. Dictionary definitions use words like “delighted, pleased, jubilant, elated, joyous.” My dog is a very happy dog, wagging his tail nearly always. He appears to be enjoying (another associated term) his circumstances. He could also be happy by nature, where he simply enjoys life or his interactions in general.
Satisfaction of course can have an according emotion, but strictly speaking, no emotion is necessary to complete satisfaction. It could be merely intellectual assent; and you can be pleased with the resulting state or outcome. Contented to me would be the lack of stressing or negative emotion, complementary to ‘satisfy’ in the context that what I have currently is enough (‘sat’).
As another person put it (on Philosophers’ Playground): “I’m satisfied that I’m happy just to be content.”