Should you run your laptop off battery power or use a charger?

"Battery chemistries have improved since the old NiCad days, but they will still exhibit some memory."

This is incorrect. Lithium ion batteries do not exhibit memory. However, the digital tools to check them can develop a memory, which leads to inaccuracies. (and is why it is recommended to occasionally cycle the battery). Cycling is hard on lithium ion, so you should try not do to it too often.

You never want to fully discharge a lithium ion battery. As I understand it, most batteries consist of multiple internal cells, and fully discharging the cells increases the chances that one of the cells will reverse polarity and ruin the battery.

Lithium ion batteries contain safety circuits that will attempt to shut down the battery before it fully discharges, but of course batteries continue to naturally self-discharge even when they are not being used, so such a shut-down battery could still fully discharge over time. So if you discharge a lithium battery to the point where it stops working, it is not actually fully discharged, but you want to recharge it soon, while it still holds this residual charge.

This is why all new lithium ion batteries ship partially charged and recommend a full charge before initial use.