What type of fire extinguisher do I need for lithium polymer batteries?
I've been doing LOTS of research on this for the past several hours on Google.  Here is what I've found:

Lithium METAL cells (coin cells like the common CR-2032 and possibly other Li-primary batteries, but I'm not as sure) truly have uncombined lithium metal in them.  Li is the lightest of the alkali metals (along with Na/sodium, K/potassium, Ru/rubidium, and Cs/cesium (caesium in the UK), it will react with ANY oxidizer, but being so light is normally isn't violent like the others.  Unlike the others, it will also react with nitrogen at room temperature, which makes fighting fires interesting (Class-D lithium extinguishers use argon as the driving gas for that reason).  The point is that if these cells physically open for any reason you WILL get a fire or at least a lot of heat.  If it's just one cell (like a coin) and not ignited yet, and you have bunch of oil nearby, you may get away with throwing it in as all the alkali (and many of the heavier alkaline) metals are stored under mineral oil.  The point is to exclude air, water, and anything containing them (like most extinguisher agents).

Meanwhile, Lithium-ION cells should not, under normal circumstances, contain ANY free metal.  The danger is that the electrolyte is often highly flammable, but you can treat it as you would, say, gasoline.  It won't spontaneously ignite in water or air, AFAIK, but opening the cell would definitely kill it.  Thus (A)BC should work fine.

Of course there's a catch--otherwise you wouldn't need the Battery Management Circuits (BMC's): over-(dis)charge can cause electrolysis which can create free metal.  If that happens, THEN you could spontaneously ignite like the Note-7's did.  A malfunctioning charger (as related above) or (more likely in production cells/packs) a short in the cells/packs/etc is the most likely cause.

It has also occurred to me that, with the only real exception (outside of factories and warehouses) being electric cars, that you won't have enough lithium around in one place to be a major threat on its own (unless holding it as it goes off, of course): rather, the danger is as an igniter lighting up regular flammable materials like furniture, carpet, etc.  In my case, this really worries me because I have several old Li-ion cells in my storage unit inside devices that are likely discharged or nearly so.  Some are single cell (phones) and will just go to zero volts.  However, there are also laptop and radio packs that have 2-3 cells in series, and these could over-discharge a cell and theoretically cause a fire.  THIS worries me but I haven't removed them yet.