Now-a-days you take their cities and they say "NO! PLEASE STOP! Give me 500 gold per turn and three of your cities!"
it depends on the AI's personality
the warmongering ones especially (Montezuma, Alexander) will just accept nothing but unconditional surrender from you, this is just from my experience playing on domination-only games.
last night's game i played, i swarmed Rome and Aztec's capital with nukes and XCOM squads paradropping in from a continent away and took their capitals and they actually surrendered a city to me each.
here's more fun from /r/civ regarding UN names:
