Agreed, but again, these are Li-Ion batteries. Li-Ion batteries are MUCH safer than Li-Po batteries, otherwise they wouldn't be in every mobile phone. I've never heard of a mobile phone bursting into flames...have you?
No, the basic Li-Ion chemistry is not much safer. There have been cases of phones overheating, and also laptop batteries. I thought the HP problems were well known. Sony too, IIRC. And, in the world of electronic cigarettes, the problem with Li-Ion batteries is well known too. Battery University is a great resource. I suggest people read there. Candlepower Forums as well.
There are some Li-Ion family battery chemistries which are much safer. Lithium-Manganese, and Lithium-Ferro-Phosphate, for example.
The Li-Poly designation has nothing to do with battery chemistry.
The primary difference is that the lithium-salt electrolyte is not held in an organic solvent but in a solid polymer composite such as polyethylene oxide or polyacrylonitrile.
The wikipedia article further mentions that the positive electrode is either Lithium Cobalt or Lithium Manganese. Of these 2, the Cobalt salt is the common Li-Ion material, and so any Li-Poly battery using such chemistry will not be inherently more or less safe than other common Li-Ion cells. There's a reason why you can find 'protected' Li-Ion batteries. The manufacturers of these (AW, Tenergy, Ultrafire, etc.) add the protection circuit to the cell. But these protection circuits can fail too. In a mobile phone, the battery is a Li-Ion, even if it's a Li-Poly configuration. The overcharge protection circuit is included in the phone's electronics. The protection circuit should also protect against over discharge, both in terms of excessive discharge, and overcurrent.
I don't use common Li-Ion cells at all (except in my phone, where I have no choice, and in fact I don't know specifically what chemistry it uses); I use LiMn, and will probably be using LiFePo in the future. I also use a Pila IBC charger. Lots of people (this is from the e-cig forums, and flashlight forums) don't trust the cheaper chargers to shut off. (Similar to a couple recent threads we've had here on el-cheapo DC-DC converters.)
Also, as I understand it, lithium fires can't be extinguished with water. So buy a good charger. If you want to be really, paranoid, there are Li-Ion charging bags.http://forcedfuel50.fatcow.com/sitebuildercontent/sitebuilderpictures/.pond/LIPOBAG.jpg.w560h446.jpg
(Found that at Super-T manufacturing, who is targeted to the e-cigarette market.)