In domestic use LiIon (Lithium Ion) batteries are, all things considered, MORE dangerous than "lead acid" batteries, not less dangerous. But both are "reasonably safe" [tm] when used properly.
The advice that you linked to above is actually titled "What precautions are needed when charging a car battery in an apartment?" and that is quite different than charging a car battery at home.
Specifically, a car battery is a one of a range of variants of lead acid batteries and contains liquid acid and while it has plugged vents and fillers it is not "sealed" in any adequate manner. Under certain conditions which are reasonably liable to be encountered in normal charging it may liberate either acid fumes or Hydrogen gas, or both. If it is charged in a car or outside it is unlikely to cause many problems.
Lithium Ion batteries when being charged do not usually liberate hydrogen or release electrolyte. Both are possible, but only if a damaged or incorrect charger is used. In exchange for not doing these things. instead they occasionally catch fire and "explode" - actually not a true explosion.
Each 18650 cell in a typical laptop battery contains the energy of about 12 high energy load '44 magnum' shells or about 24 "standard" .44 Magnum rounds, and that's just the electrical energy. (The energy from combustion can exceed 300 kJ, or over a hundred .44 Magnum rounds!). An entry level netbook has 3 such cells and a top notebook PC may have 9 or even 12 of these. This can be released in about 10 seconds with flame and 'lots of heat'. The standard industry term for this, somewhat tongue in cheek is "Vent with flame". The fact that a standard industry term exists indicates that it's a well known problem. The "melt down" mode can be triggered by charging to too high a voltage, discharging excessively and then charging normally, charging at an excessive current rate, puncturing them, or in selected cases* giving them a slightly harder than usual knock which causes internal parts to short together and - Wow!. (* This is obviously a manufacturing fault but has happened in a number of independent cases due to too tight manufacturing tolerances and substandard assembly).
All that said - given how many there are, LiIon batteries have proved to be very safe considering how many there are and how they are treated. Despite the horror stories I have never seen a real world "vent with flames" event.
Added 7 years on: I've now seen one. An old netbook with a dead LiIon battery pack was left connected to a power supply for several days. One evening it sudeenly burst into magnificent melt-down. Flame, smoke, ... - urgent defenestration of the battery save the netbook - and the couch was "not badly damaged". Had we not been in the room we'd have had a house fire.
Lead acid batteries do not have an equivalent failure mode to "vent with flame"
However, drop a spanner, vacuum cleaner metal suction tube or similar across the terminals of a car battery and you'll get immense energy release. The battery may be damaged by such treatment but usually won't explode. Charge them too fast or too long and Hydrogen gas will be produced and WILL leak out and can form a flammable or explosive mix in confined spaces. Battery acid from other than fully sealed lead acid batteries seems to be special Houdini grade - skilled at escaping in many unexpected instances. If you carry the battery and your arms itch, Wash them NOW. Next time you wash your clothes small holes with brown edges may appear(Ask me how I know). Charge them inside at above gassing point and you may be sorry. I was :-).
BUT Lead acid are also very safe as long as they are used as intended. Charging a wet plate lead acid car battery "inside" at home is not included in "as intended".
Some excellent related links, recommended by Nick Alexeev:
MPower UK - Lithium Battery Failures
SE EE - Why is there so much fear surrounding LiPo batteries? - good question and eleven good to great answers.
A good place to answer many battery questions is Battery University. They discuss a range of things on lead acid and many other battery technologies at various places on their site.
You could look at their Charging lead acid page to start but for completeness should have a look at the site from the top level to see what else is relevant.
Note that battery capacity is stated in Ah (ampere hours) or Wh (Watt hours).
Wh = Ah x Battery_Voltage.
A "rule of thumb" is that lead acid batteries should not be charged at above the C/10 rate. but in fact far higher rates are acceptable in many cases if due care is taken. For 2.3 Ah C/10 = 230 mA and for 250 Ah BATTERY C/10 = 25A.
Other concerns are type of lead acid technology, which affects final voltages. You need to read up on float voltage, topping cycles (not related to Space Shuttle main engines) and equalisation. Those aspects and more are covered on the battery university site.
From the above 'Charging lead acid' page:
Figure 4-4: Charge stages of a lead acid battery
The battery is fully charged when the current drops to a pre-determined level or levels out in stage 2.
The float voltage must be reduced at full charge.
During the constant-current charge, the battery charges to 70 percent in 5–8 hours;
the remaining 30 percent is filled with the slower topping charge that lasts another 7–10 hours.
The topping charge is essential for the well-being of the battery and can be compared to a little rest after a good meal.
If deprived, the battery will eventually lose the ability to accept a full charge and the performance will decrease due to sulfation.
The float charge in the third stage maintains the battery at full charge.
The switch from Stage 1 to 2 occurs seamlessly and happens when the battery reaches the set voltage limit. The current begins to drop as the battery starts to saturate, and full charge is reached when the current decreases to the three percent level of the rated current.
A battery with high leakage may never attain this low saturation current, and a plateau timer takes over to initialize the charge termination.
The correct setting of the charge voltage is critical and ranges from 2.30 to 2.45V per cell. Setting the voltage threshold is a compromise, and battery experts refer to this as “dancing on the head of a needle.”
On one hand, the battery wants to be fully charged to get maximum capacity and avoid sulfation on the negative plate; on the other hand, an over-saturated condition causes grid corrosion on the positive plate and induces gassing.
To make “dancing on the head of a needle” more difficult, the battery voltage shifts with temperature. Warmer surroundings require slightly lower voltage thresholds and a cold ambient prefers a higher level.
Chargers exposed to temperature fluctuations should include temperature sensors to adjust the charge voltage for optimum charge efficiency. If this is not possible, it is better to choose a lower voltage for safety reasons. Table 4-5 compares the advantages and limitations of various peak voltage settings
See above page for table and very substantially more information.
Best Answer
You are entirely correct. The issue comes down to the fact that it's pretty easy to overcharge a lead-acid, but normally you try not to "over-discharge". So you have to be careful about the last stages of charging. This is not (exactly) true for discharging, but that is only true because you don't want to completely discharge lead-acid anyways. Unless you have a deep-cycle battery, you don't want to pull more than about 50% of the available charge out when discharging. If you do, you'll severely reduce the battery life.