For the Epicureans happiness is a life of pleasure, because the highest good is pleasure. Although they qualify that by saying that of all pleasures the highest is lack of pain. So they really seek a life without pain (especially mental).
The jury is our about what most of us seek, as you probably know there are a lot of modern empirical studies on that. But the point is that lots of people think pleasure is a fundamental component of happiness, which it isn’t. That’s why, say, buying new stuff all the time (hedonic treadmill) doesn’t result in an increase in self-reported happiness. On the contrary, it decreases it.
Finally, remember that for the Stoics the only good is virtue (or a good character), but pleasure is “preferred,” so long as one doesn’t seek it in unvirtuous ways.