Sorry to answer by reference to Wikipedia, but, well, I think the answers are all there.
We have ancient philosophical schools.
We have popularisations of what those ancient philosophical schools were about, in education and in all-round educated discourse.
We have people repurposing those popularisations, to express commonplace attitudes.
To the extent that the meaning differs from antiquity, that’s the result of popularisation.
The Stoics taught that destructive emotions resulted from errors in judgment, of the active relationship between cosmic determinism and human freedom, and the belief that it is virtuous to maintain a will (called prohairesis) that is in accord with nature.
Later Stoics—such as Seneca and Epictetus—emphasized that, because “virtue is sufficient for happiness”, a sage was immune to misfortune.
The word “stoic” commonly refers to someone indifferent to pain, pleasure, grief, or joy. The modern usage as “person who represses feelings or endures patiently” was first cited in 1579 as a noun, and 1596 as an adjective. In contrast to the term “Epicurean”, the Stanford Encyclopedia of Philosophy’s entry on Stoicism notes, “the sense of the English adjective ‘stoical’ is not utterly misleading with regard to its philosophical origins.”
And indeed, if you read Epictetus, the stuff of pop stoicism is there: you should only care about the stuff you can control, there’s no point being bothered by the stuff that’s outside of your control. So you don’t let pain, pleasure, grief or joy from those things run your life. It’s just that there’s a calm—at times, a defiance, even a joy, in Epictetus not bothering with the stuff you can’t control. And there is a sense of purpose in the stuff you can control, which does give you joy.
The pop version of stoicism, as exemplified by the British stiff upper lip, doesn’t have that; it’s more absolute. But you can see how you would jump from one to the other in the 16th century.
Now Cynicism (philosophy):
Cynicism is a school of Ancient Greek philosophy as practiced by the Cynics. For the Cynics, the purpose of life was to live in virtue, in agreement with nature. As reasoning creatures, people could gain happiness by rigorous training and by living in a way which was natural for themselves, rejecting all conventional desires for wealth, power, sex and fame. Instead, they were to lead a simple life free from all possessions.
That’s not small-c cynical.
By the 19th century, emphasis on the negative aspects of Cynic philosophy led to the modern understanding of cynicism to mean a disposition of disbelief in the sincerity or goodness of human motives and actions.
Oh, OK. So what gave people that idea?
The ancient Cynics rejected conventional social values, and would criticise the types of behaviours, such as greed, which they viewed as causing suffering. Emphasis on this aspect of their teachings led, in the late 18th and early 19th centuries, to the modern understanding of cynicism as “an attitude of scornful or jaded negativity, especially a general distrust of the integrity or professed motives of others.” This modern definition of cynicism is in marked contrast to the ancient philosophy, which emphasized “virtue and moral freedom in liberation from desire.”
Ah. So too many anecdotes in the 1800s of Diogenes of Sinope flouting conventional morality to point out hypocrisy. And not enough justification of what Diogenes’ point was, behind his antics—that there was true virtue to be found out there, it just wasn’t to be found in conventional pieties.