That's not quite right. but you have the right idea
AI gets chieftain level happiness, that's what the often referred to handicap xml file is actually describing chieftain settings -which also applies when humans play chieftain difficulty (like happiness bonus, unit support etc)
On Chieftian difficulty, the AI gets chieftain level happiness but are hit with negative handicaps like slower build speed. And are less agressive.
On Prince, the AI still have chieftain happiness, but their handicaps are now even.
Above that, they get a bonus.
So no, they don't play on chieftain difficulty all the time, they play on the difficulty you set the game to.
And Prince is still the even level.