what I don't know is does that damage the battery to run it completely dry like it does with lead Acid
Most lithium battery packs for Laptops etc have a circuit board embedded in them with a "supervisory" chip and some switching transistors. The chip monitors the contacts and when it sees a charger it turns on the charge transistor and turns off the supply transistor, this allows the pack to charge. When the charger is removed the chip turns off the charger transistor and turns on the supply transistor so the battery pack can power whatever unit it is built to power (laptop, vacuum, etc)
The supervisor chip also monitors the battery voltage and won't let it go below a preset limit. As an example the S-8254 chip, which is pretty common, won't let a 3.6V lithium battery go below 2.0volts. It will shut off the supply transistor if it detects the battery has dropped this low. This protects the battery pack. It does a similar check/test for overcharge, if the battery reaches 4.4V it turns off the charge transistor.
A quick Google search for "Abilic Inc" "S-8254A" should provide anyone that wants to read further a free datasheet for this IC and a good primer on how some battery packs work.
So the life of a 3.6V lithium is between 2.0V and 4.4V with a typical resting point of 3.6V
Don't confuse this with lead acid technology, just because the battery cuts out at 2.0V doesn't mean that you only get (3.6-2.0)/3.6=44% of the charge. You get closer to 90%
There are two big caveats to lithium battery packs. If the batteries get depleted below 1.1V they are dead and cannot be recharged with a conventional charger, Once the chip loses power both transistors turn off and then there is no way to charge it. It's a chicken/egg problem, the chip doesn't have any power so it can't turn the transistor on to allow a charge to occur and because you cannot charge the battery you cannot get any power to the chip. Neat obsolescence trick eh?
Lithium batteries can lose between 5% and 10% per month so if your batteries sit too long they turn into bricks. I had a customer that bought brand new electric 4x4 ATVs for his cottage. Parked them in the fall and didn't visit the cottage again until the spring. The ATVs were dead and refused to charge when hooked back up to the charger. He had to go buy new battery packs for both.
Sometimes it just pays to KISS.