What is a non-lethal kill?

They're now labelled as "takedowns", so that's one problem solved.

As you suspect, a takedown is when you either use the "takedown" action, or "kill" someone using non-lethal damage. That is, via stun or any weapon that states it does non-lethal damage in its description.

Non-lethal weapon list:

• Unarmed
• Sleeptite Atomizer
• Electric Truncheon
• "The Help"
• "Jack the Nipper"
• Jolly Brolly ^preorder
• The "Nighty Night"
• The "Sledger"

While the Electric Truncheon claims to do very light lethal damage, it's far more likely to stun someone unconscious before it kills. I don't know whether this is a discrepancy in its description (i.e. it actually does heavy non-lethal damage), nor whether it's possible to get a takedown with the heavier stun weapons before they kill.

Unconscious people eventually wake up and walk away when you're not looking.

This is simply flavour text for the fact that bodies despawn and NPC names are reused. There is no equivalent explanation as to how you can meet people you definitely killed before. Killing or performing a takedown again on the "same" NPC still increases the counter.

According to Reddit's article about non-lethal weapons, it seems like it would be a "kill" that is performed using a non-lethal weapon, the aforementioned thread contains a few discussions of weapons of that sort.