Purpose of the slot at the base of the drill’s handle

The electric screwdriver can sometimes fit where a bulky drill cannot, e.g., electrical boxes. Apart from that, the drill is a fine screwdriver (assuming it has a clutch, of course, but I have never seen a battery-powered drill that does not).

Don't get rid of your corded drill. Battery-powered drills, especially the cheap ones, are not powerful enough for serious drilling such as a 5/8-inch hole through a 4x4 or any hole through metal.

Most drills (all the ones I know about) have a physical switch that connects to a relay that actually completes the circuit that will get the drill to spin (like in this circuit). So unless there is a display or something that requires constant power (like if those LEDS had their own switch) the batteries should hold their charge as long they are chemically able to do so. The NiCad battery may have issues maintaining its charge compared to the LiIon battery but you should not be able to tell the difference after one day. Even though neither of them should be dead after one day.

The LiIon battery may have been close to dead without you noticing since they tend to put out a constant voltage even while at low power levels; however, the NiCad should have showed some signs of being low.

Regarding the weather, NiCad should be able to tolerate the cold fairly well. LiIon on the other hadn can have damage to the cells from really cold temperatures but i've left tools out overnight in freezing weather (0F-10F) many times and once they warmed up a bit they worked fine (ignoring the fact I probably lowered amount of recharge cycles they had left).

All that being said I also think its really unlikely that both batteries/ drills (or any combination of the 4) have issues... So other than the fact you angered the pixies living in your tools by leaving them out in the cold, I would guess the long term storage in a place with large temperature changes shortened the capacity of your batteries.

PS The reason LiIon batteries are stored at roughly 40% capacity is that part of the trade off of being able to have a bunch of small charge cycles going up and down a lot (like cell phones) is the fact they don't do too well at the extremes (0% & 100%) because one way chemical changes are more likely to happen (lowering their capacity and cycle life).