For runs like "Kill the bile titan" does the run end upon the text saying you've killed the bile titan or do you need to finish the mission?

Also if you do need to finish the mission does that mean extracting? or can you simply just run out of lives which ends the mission too?

the run ends when your mission ends. You need to 1) finish the objective and 2) finish the mission. This can happen by extracting or by running out of reinforces. The time used is in-game time (at the end of the mission your time will be displayed on the top right)

Ok thanks for the info!