Why don't all devices use this? It adds cost and complexity. Is their any other reason for not doing something?
Seriously, I'd say that there are plenty of options and implementations for this. Having two equal batteries doesn't make much sense, so often the second is used for emergency or limp-home power. For instance, your PC has a RAM retaining battery on the motherboard for when you loose power. A laptop often gives a "Low battery" warning, at which time you're welcome to reduce power however you can.
I think that your statement that 'batteries work best if they are used until they are completely drained, and then recharged.' is a little broad. This is more the case for Nickel-based (NiCd and, to a lesser extent, NiMH) chemistries. Lithium Ion cells don't suffer this memory problem. In fact, their lifetime improves if you avoid deep discharges. See this page from BatteryUniversity.com for reference.
There are a couple of options for doing more intelligent power management in your own devices.
The simplest is an ORing diode on the power supply. If all you want is a hot-swappable power supply and you have a bit of leeway for your inputs, you can connect backup battery to the anode of a diode, and connect the cathode to your main battery. When the voltage of the main battery dips to 0.7V less than your backup (Or is removed), the other battery kicks in. Be careful of leakage current into the backup battery, it might overcharge it.
Alternatively, you can use a power mux IC like the TPS110. This lets you select your input independently (or dependently, if you prefer) of the input voltages, instead of always using the higher supply.
Finally, Linear Technology incorporates what they call "PowerPath" controllers into their battery charging ICs. I've used their LTC4011 which seamlessly transitions between battery and external power, and charges the battery while running off of the external power.
Any two of voltage, current, and power can be specified and the third inferred. Which two are listed in the datasheet depends on what is customary to talk about for that class of device, what the datasheet writers think is more relevant, or what matters most to the design engineers that give the specs to the datasheet writer. Sometimes with lesser datasheets you get the parameters marketing wants to push because they think their device is better than the competition in that area. It varies.
If a devices is rated to require 92 Watts at 12 Volts, then you can infer it draws 7.7 Amps at that operating point. Again, you can compute any one of these numbers from the other two.
Be careful what the power spec really means. There are two different specs that have units of power for Peltier coolers. One is the electrical power the device will use, the other is the thermal cooling power. Both are relevant to different parts of the design. Possibly only one of these is listed and a efficiency spec is provided so that you can determine the other. With Peltier coolers, efficiency is a strong inverse function of input current. There should be either a equation or a graph in the datasheet showing the relationship.
As always, it's important to read the datasheet carefully. If you find a spec that doesn't make sense, come back and ask about it specifically. Otherwise there is too much general stuff to get into here.
As a separate issue, 92 Watts is way way way too much power to expect from a few "pencil cells". Go read a datasheet for any ordinary AA battery, paying close attention to maximum current and derating at high current levels.
Best Answer
There are multiple reasons that all come into play when dealing with the power requirements of a new laptop design. The main goals are to have the longest battery on/standby time possible while still being light, small, reliable/resilient, and quick to charge.
The battery must be able to provide enough power for the laptop to be on for a couple hours. This power requirement varies from laptop to laptop based on how much power the device uses. Here are the battery and power supply specs of a few laptops I have laying around.
Now because the batteries have different voltages, capacities, and technologies they need to be charged differently. This is where the different power supplies come in, they need to be able to power the laptop, and be able to charge the battery at the same time, without damaging or overcharging it. You can destroy some batteries, even causing fires or worse by charging them with the wrong type of charger. Take a look at Battery Chargers and Charging Methods for more information. I think most laptop batteries are lithium-ion, but I don't know that for sure.
So for some charges, there is a lot more to do than just providing 12volts with up to a few amps. Depending on the laptop design, the battery charging circuit (the circuit that prevents the battery from overcharging or being charged too fast, etc.) may be located inside the laptop itself or in the charging unit. If the only thing the power supply does is to provide a voltage around 12 volts, then you may be able to run it off your car cigarette lighter. But I would not recommend this since the voltage will vary greatly. Vehicles, especially older ones, do not provide a constant 12volts. As for the amperage, a cigarette lighter should provide more than enough amperage, and if not you would most likely just blow an accessory fuse. However if you laptop's current power supply is around 9volts or less, you could probably rig up a DC to DC power regulator to provide that voltage from the vehicle.
As for the plug that they use to connect to the laptop, for the most part it doesn't matter. Sometimes chargers provide multiple voltages, and/or have feedback that requires more than two conductors but for the most part they only need 2. The plug you referenced in your question, "yellow tip EIAJ-04 power jack," is just a generic coaxial plug that provides two conductors (inner and outer) It is used by some different laptops. But a reason that they all don't use the same plug is so that you can't hurt your laptop by pulling in the wrong adapter, and I'm sure there is a bit of hope that if your power supply gets broken or lost that you will buy another one from them since it may be hard to get a generic one.
Keep in mind that the plug its self is not really important, it's the power supply that connects via the plug that matters.