As principle behind this think about your old light bulb. One emergency method of lighting a fire used to be to break the glass of the light bulb in your flashlight and burn the filament, no more no less than the way its done with a strand of steel wool for example.
Now, when you put the filament between positive and negative there will be some resistance in the filament. This "friction" is what heats it up to incandescence and proceeds to burning it out.
Obviously the energy dumped by the battery is wasted in heat by the filament and the circuit is left open once the filament burned. This is very different than shorting an AA battery using something rather thick, like a paperclip for example. Simplifying: the paperclip, being relatively thick, will not offer enough resistance to dissipate that energy in heat and the battery will be the one suffering for it. For a wire of an X material, the thinner the more resistivity it gets, steel wool is very thin in the finest grades, a light bulb filament is really long and really really thin. When you talk about materials the principle is the same: electrical resistance of one material will be different than the resistivity of another (less resistance = less energy dissipated in heat to turn in incandescence useful for us and more energy the battery has to suffer) Copper has a very low resistivity and high conductivity, that's why we make electrical wires out of it, and we don't want those to heat up too much, steel on the other hand is obviously conductive but offers quite some resistivity, and when the filament is thin will just burn and melt off ( thats what we are looking for). Carbon and graphite are even worse than steel in fact, one of the first light bulbs demonstrated, used carbon filaments
Now, lets take one of the videos you linked as example: it uses a piece of pencil lead being burned by jumper cables connected to a car battery. In principle no different than a car cigarette lighter but with one big difference in practice: in your car the circuit and your battery are protected by fuses, these, as much as your steel wool filament, are set to "burn out" when there's the chance that the circuit will end shorting out your battery. In the video you are just assuming that the pencil lead will burn out opening the circuit before shorting the battery and that's not always true, the smoke might end coming out from the wrong place. You want to use your car battery? Pull the wires you need from the lighter plug or pull a signal light and break that, at least there will be some fuses in the circuit. With a car battery you wont get shocked, its just about 13V delivered, but the amperage that the battery is capable of delivering is very high, while you need to dissipate all that energy or protect the battery in some way when in a circuit, dry skin offers too much resistance for the amount of volts to push all the amperes the battery is capable through an healty body. If you touch the wires to create sparks instead, both the wires and the sparks can leave burns (and that's also not good for the battery).
The cellphone battery example: these days many have short circuit protection ( IEC62133, since 2012, is one of the normative used as a standard for these things, it does vary from country to country though). If you go buy generic lithium batteries you find that they are not all created equal, some are protected, others are not, some are suitable for one thing while some are suitable for something else. When it comes to protection circuit of a cellphone battery the threshold can be set so low that some steel wool will trip it, no juice delivered anymore, depending on the protection system used this can be a temporary switching off or permanent (you might lso find interesting reading about Intrinsic Safety and battery design). A generic, unprotected, el-chapo offbrand lithium battery bought on Ebay can mean trouble in any application. Think that when it comes to these batteries even making your own flashlight requires that you pair a lamp compatible with the cells you will be using so you cant just say that a lithium battery will explode if you use that way, but at the same time just mind that unprotected lithium batteries are not that safe if used outside their specs, so when it comes to lighting a fire (in the fire pit and not on the battery) these might not be the best tool for the job. They are also not cheap so I wouldn't use them just for experimenting, they are more useful in the phone.
In one of your examples is mentioned a "brillo-pad": given this is basically the common name used for whatever steel pad used to clean a pot be aware that some are way too thick to give the results you want. Stick with steel wool.
The "jail" method that uses gum wrapper or a thin piece of the foil in cigarette boxes works because the material is very thin. If you were to use the tinfoil used in kitchens chances are its too thick (tinfoil thicknesses vary)
At the end, the finest steel wool is a material that you know works with everything, its so thin and its resistance so high that there's no particular risk involved and an AA battery can light it just fine, a 9v is just much easier to use since the terminals are one near the other. Just don't store the two together or you will light a fire in your pocket.
And talking about storage, many kept spare batteries in the junk drawer without worrying about possibility of shorts and cabinets on fire, and that is more hazardous than lighting a camping fire with steel wool and a 9V battery.
You were asking about which materials can be safe with which battery. That can go a bit on experience, experimentation or crunching numbers (a resistance formula R = ρL/A where R is the total resistance, ρ is the specific resistance of the stuff the wire is made of, L is the wire's length, and A is the thickness of the wire, will tell you what to compare to the battery specs, however noone will bother calculating this ;) ) You don't need to risk anything, once you know the principle you will have a clear idea of what works or not.
One of your links (lifehacker) mentions getting sparks from shorting the terminals... that's not what the steel wool+battery does, you dont try to create sparks to light stuff up, you are literally heating and burning up the steel wool itself, it catches fire.
If you want to be really safe you can always make your battery from a lemon and use that to light your steel wool, it wont explode and it will be still fit for eating afterwards. Add a small ball of lint with vaseline if you like.