There is some misunderstanding about this that comes down to two issues:
1. There is no such thing as a safe battery. They can all get very hot if shorted out and left like that, especially in some kind of confined space, as that will increase the heat.
2. AW batteries are safer for use in mods because they are less likely to explode, and to do that in front of your face.
I would only buy AWs for my mods now, but that does not mean I regard them as safe. There are no safe rechargeables. Maybe the battery info page needs to make that clearer - a mistreated battery is still a danger. However, as stated, some are less likely than others to blow up in your face - which was what the issue was when we decided to give advice.
And it has always been the case that any mod should have gas vents, a master off switch, and preferably some sort of dead short aka short-circuit protection. Lithium batteries are not intrinsically safe. Some are less likely than others to blow up in your face, which is mainly what the advice is about.
All rechargeable batteries need to be treated with care, they aren't really in the fridge or TV class of product where you buy it and forget about it, and don't have to worry about the item having safety issues. As far as we know, Li-Po cells are the most vulnerable, through Li-ion, then Li-FePo4, to Li-Mn as the 'safest'. But the term 'safe' is relative when referring to lithium batteries. Protected Li-ion cells are fine if the protection circuit is in perfect condition; but it may not be.
All things considered, AW cells are voted by those who know about these things as the 'safest' to put right in front of your face when fresh off charge and to pull a relatively high current through. As regards other measures of safety, i.e. leaving alone in the house when on charge, or left alone in a confined area in a device that may not have safety circuits or even the vital master on/off switch that we have been banging on about for years - well, probably no battery is safe in those circumstances. Mods need to be built with the idea that things will go wrong, with layers of protection, so that when one protection fails, another (that may well do exactly the same thing) kicks in. The idea that "You don't need Protection B because it already has Protection A" is just simply wrong, as any engineer will tell you. Things fail, and protection devices are like anything else - they fail (or don't get used due to operator error).
A mod needs gas vents, a master on/off switch, and probably some sort of protection circuitry as well. If you want to buy one without those things, it's up to you. You should recharge your batteries in a LiPo charging sack or in a vented cake tin, and only when people are in the house and awake; and you should not jam it in your pocket along with keys, coins and pens, without the master switch set to off, and then leave it unattended (and especially not have spare batteries that can contact anything metal).
An APV is just not a buy and forget device. The problem is, people think it is.
As regards giving some sort of verdict on the incident discussed here: that is impossible without more information. For example: is the case in the photos a soft carry case, or is it Eastmall's USB Battery Box? Did it have some metal cartos or tanks in there too? Does the Indulgence have a master on/off switch? And so on.
[edit]
I edited the Battery Info page to make it clearer that the safety issue under discussion on that page is the exploding in your face issue, and to make it clear that there are no safe lithium cells.
And to labor the point even more: there are no safe lithium cells. Leaving an APV with no safety features in your coat pocket, or expecting that there won't be a fire if your spare batteries short out, is unrealistic. These things have a known risk, and sensible people try to minimise the risks of them shorting out or an APV firing in a pocket or purse.
A single battery, or a pair of batteries, won't catch on fire unless they are shorted out somehow. The fault is in the situation that allowed the short-circuit to occur, not in the batteries.