The best way to characterize things is that lithium ion batteries have a certain number of full cycles before they start to show reduced capacity. With modern batteries, it's a few hundred full cycles, which is discharging from 100% to 0% and charging back up to 100%. A particial discharge is just that--not a full discharge--and so an li-ion battery is going to last longer if it's not seeing a full discharge every day.
However, it's simply not true that an li-ion battery is "permanently damaged" when it's discharged to 0% and that it "might never power on again." Anybody who has routinely discharged their devices to 0% should know this. And, discharging below 40% just means that it's going to require more charge to get back to 100%. But it means nothing more than that.
Yes, it's a good idea to avoid deep discharges, because that means your battery will last longer before it starts to suffer from reduced capacity. So, sure, charge it when you can. But you're not risking destroying your battery or having your device never start up again just because you let it discharge to 0% on occasion.