The goal is almost never to kill or capture. That’s not why you’re there in the first place, is it? The goal is to achieve some effect. If you can do that without killing or capturing then all the better. If you narrowly focus on killing and capturing then you may be missing other options which are easier. That’s what Sun Tzu is reminding us, and it’s why a modern tactical checklist includes an item like ‘what effect do I want to have on the enemy?’ so you don’t just go straight to kill or capture.
I'm not sure if it was also Sun Tzu or perhaps Marcus Aurelius who also mention that the most vulnerable state an army can be in is when it routs (this is borne out historically, as most battles with very heavy casualties on one side suffer them during a rout). So if you allow the enemy a gap to flee you have options to either spare them or wipe them out depending on your goal as a commander.
They got so good at this tactic (which is one of the hardest to pull off, even in modern combat) that they had a special word for the manoeuvre, but my Google-fu has unfortunately failed me in trying to dig it up.
Across the few Khan and Mongol histories I've read the tactics are there, but often end up with different names. Like the only tactic that springs to mind right now is to spread out a march to stay low profile during movement. They would call that "moving tree formation", or "moving bush", or "waking tree", or some other related variation from similar root. A quick search brought me to [1], which calls it the new-to-me "bush clump tactic". This book calls your feigned retreat "Luring into Ambushes".
Translation seems to bring a lot of variation in these supposed names. So I end up entirely unsure what they really called them. :)
[1] http://deremilitari.org/2014/06/the-art-of-war-under-chinggi...
Thanks for doing the digging up on all the names :) It really is interesting how the tactic has been used and adapted over time and place, and how the names necessarily would be different depending on who you ask. I guess calling it any of the above is totally valid XD